Benchmark Tests (Day 10)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

This rectangle's grassy area is 6.75 sq. in. Its length is 13 more than its width. What is its perimeter?

Actually, I wasn't quite sure whether to classify this problem as Geometry or Algebra I. It involves the area of a rectangle so technically it's Geometry, but after the trivial fact of giving the area of a rectangle, it's essentially just Algebra I. Let x be the width so that x + 13 is the width:

x(x + 13) = 6.75
x^2 + 13x = 6.75

In theory, every single quadratic that Pappas asks us to solve can be so done by factoring. After all, the answers to Pappas questions are whole numbers, and any quadratic polynomial with rational zeros must be factorable.

But that 6.75 discourages us from trying to factor. We could multiply straight through by 100 to clear the decimals, resulting in large coefficients. If we recognize 0.75 as 3/4, we might realize that we could just multiply by 4 instead of 100, giving us something that's easier to factor.

Our students would most likely use the Quadratic Formula to solve this equation. But this is an interesting one to try completing the square. Half of 13 is 6.5 and 6.5 squared is 42.25:

x^2 + 13x + 42.25 = 6.75 + 42.25
(x + 6.5)^2 = 49
x + 6.5 = +/- 7
x = 0.5 (discarding the negative value of x)

The length is 13 + 0.5 = 13.5, and the perimeter is 13.5 + 0.5 + 13.5 + 0.5. Therefore the perimeter of the rectangle is 28 -- and of course, today's date is the 28th.

Today I'm posting the Benchmark Tests. Perimeter and area do appear on this benchmark, and so today's Pappas question might help us.

Here is the Blaugust prompt for today:

Observe yourself!  Record your lesson using your phone in your pocket and use it to reflect

Well, you already know that I have very few photos or videos of my classroom to post. Anyway, today's a great day to sit out Blaugust anyway. Since I'm posting a test today, this means it's time for another traditionalists post.

There is a recent New York Times article written by Barbara Oakley, titled "Make Your Daughter Practice Math. She'll Thank You Later." In this article, Oakley takes the traditionalists' side of the debate and implies that traditional math will especially help girls succeed in math. This explosive article has drawn hundreds of comments from both sides of the debate.

When I first read this article, I knew that I had to blog about it. But at the time that the article was published (August 7th), I was still tied up with finishing the last two chapters of Van Brummelen, and my next scheduled traditionalists post wasn't until today, a full three weeks later.

Meanwhile, our favorite traditionalist, Barry Garelick, has also refrained from blogging about the Oakley article -- until this past weekend, that is. It's almost as if Garelick already knew that I'm making a traditionalists post this week -- so by waiting until now to write about it, he'd be sure that I would link back to his blog.

Anyway, here's what Garelick has to say about the Oakley article:

https://traditionalmath.wordpress.com/2018/08/26/rebuttals-heralded-as-truth-dept/

Barbara Oakley wrote an op-ed in the NY Times, that went against the current faddish and prevalent thinking that math taught in the traditional manner kills all desires to learn math. She advocates learning the foundational aspects of math through practice (derisively referred to as ‘drill’) and cites her own experience learning Russian at the Defense Language Institute–one of the best language learning facilities in the world.

But then he moves on to discuss a rebuttal to Oakley at another blog:

A recent blog post offered a rebuttal to this, which is not unusual if you read the comments to Oakley’s NY Times piece.  People do not like being told that traditional methods of math teaching have merit–particularly when the current “narrative” about traditional math teaching is that it doesn’t work and turns students off of math.  It is portrayed as nothing but drill, rote learning, no connection between concepts and no understanding. Also, they tend to make questionable statements such as this: “Whereas [Oakley] says we have foregone drill and practice for conceptual understanding, our problem in the United States is understood by learning scientists to be precisely the opposite.”

The author of this blog appears to be Emma Gargoetzi, a former New York high school Geometry teacher who is no longer teaching in order to pursue a Ph.D. Even though Emma isn't a member of the MTBoS (much less a Blaugust participant), her post has drawn comments from some famous names -- including Wendy Menard and the former King of the MTBoS, Dan Meyer.

And there's one more famous commenter in the thread -- Barbara Oakley herself. Garelick has already quoted Oakley's response, and I'll do the same:

Barbara Oakley:

I learned Russian at the Defence Language Institute–one of the best language learning facilities in the world. The learning practices there included plenty of drill and yes, rote memorization, along with conversation and application. When I applied those same techniques to learning math when I started on trying to learn remedial high school algebra at age 26, it worked beautifully. As well it should, since the Defence Language Institute approach involves virtually all aspects of what we know about how to learn a language–or any subject–well.

My own daughters received plenty of drill practice of math in the ten years I had them in Kumon mathematics, from ages 3 to 13. According to the precepts of the above blog post, my daughters should hate math. Instead, their early wobbly dislike for math turned into expertise and enjoyment–just as with a child who spends not-always-fun time with piano practice can grow into an adult who treasures her ability to play the piano.

The vast majority of my colleagues in engineering are from countries that emphasize rote approaches to learning math. Their early rote training certainly didn’t kill their desire or ability to excel in analytical topics. Virtually every subject we know of, from language learning to playing an instrument, to learning in math and science, proceeds from a basis of solid mental representations developed through practice and procedural fluency that minimizes cognitive load while maximizing access to neurally embedded information. (See Anders Ericsson’s seminal work on the development of expertise.)

Many schools in the US been strongly discouraged by previous policies of the NCTM from encouraging students to develop any type of procedural fluency or deeply embedded sets of concepts, such as multiplication tables. (Yes, even memorization of multiplication tables helps children to develop pattern as well as number sense.) I still remember the student who came up to me after flunking a statistics test, saying “I just don’t see how I could have flunked this test–I understood it when you said it in class.” We’ve gone so overboard with the value of conceptual understanding that students think it’s the golden key–they don’t need to practice to build, maintain, or enhance their “understanding.” And of course, what a student thinks they understand is often only a glimmer of the real understanding that comes from plenty of interleaved practice. In reality, procedural fluency and understanding proceed hand-in-hand. See Rittle-Johnson, B, et al. “Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics.” Educational Psychology Review 27, 4 (2015): 587-597.

Note that the above blog post cites only one-sided articles in support of the author’s message–no mention is made of the researchers and results cited in the original op-ed, or other meaningful, solid research such as the following, that rebuts the author’s assertions. See, for example:

Morgan, PL, et al. “Which instructional practices most help first-grade students with and without mathematics difficulties?” Educational Evaluation and Policy Analysis 37, 2 (2015): 184-205.
[Morgan notes that music, movement, and manipulatives are fun, but the basics, with explicit instruction and plenty of worksheet practice, are best for struggling math students.]
Geary, DC, et al. “Introduction: Cognitive foundations of mathematical interventions and early numeracy influences.” In Mathematical Cognition and Learning, Vol 5: Elsevier, 2019.
Geary, DC, et al. “Developmental Change in the Influence of Domain-General Abilities and Domain-Specific Knowledge on Mathematics Achievement: An Eight-Year Longitudinal Study.” Journal of Educational Psychology 109, 5 (2017): 680-693.

Neuroscience and cognitive psychology are doing a great deal to advance our understanding of what is necessary to excel in a given subject. I would hope that educators in mathematics would open their eyes to new and relevant insights from these disciplines, and realize that their desire to always make their subject fun (something that teachers of virtually every other subject realize just isn’t possible), results in disempowering the very students they mean to help.

And after Gargoetzi acknowledges Oakley's response, the journalist cites one more article to support her side (the traditionalist or "practice helps make perfect" side):

Barbara Oakley:
(Another good reference is that of Dr. Wu at Berkeley, “”Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education.”) 

Dr. Wu at Berkeley -- that must refer to Hung-Hsi Wu. That's the same Hung-Hsi Wu whose articles on Common Core Geometry I've been linking to on this blog the past four years! Here is a link to the article from the Wu site itself:

https://math.berkeley.edu/~wu/wu1999.pdf

And not to be outdone, Garelick has also drawn a big name commenter -- the traditionalist Sandra Stotsky, one of the earliest Common Core critics:

Sandra Stotsky:

It’s not going to be easy to address Oakley’s concerns.

https://nonpartisaneducation.org/Review/Articles/CommonCoreCollaborators.htm

Here's what I have to say about Oakley's response. Oakley mentions both foreign languages and musical instruments as things that require repetitive drill and practice to learn, so she wonders why math should be any different.

Well, here's the difference between Russian/piano and math. Students tend not to learn something unless it's easy, fun, or high-status. Both foreign languages and musical instruments confer high status upon their learners. Students want to be able to read a Russian menu or a musical score (or both to play and sing Tchaikovsky's opera Queen of Spades). On the other hand, students don't generally want to learn math. Therefore, learning math must become either easy or fun -- otherwise students will refuse to learn it.

Like most of her fellow traditionalists, Oakley assumes that just because you hand your daughter a math problem set, she'll complete the worksheet and work math. Instead, there's a great chance that she won't even start problem #1. Oakley's reasoning about learning Russian, piano, or math is sound provided that the students actually do the work to learn any of them. But your daughter won't learn any math at all from an assignment that she leaves blank.

It's interesting that Oakley would cite Wu here, considering that Wu appears to be pro-Common Core while most traditionalists (especially Stotsky) are anti-Common Core. Then again, in his article that I linked to above, Wu himself warns us not to make a false dichotomy between basic skills and conceptual understanding.

By the way, I'm very surprised that not only has traditionalist SteveH not posted in response to this Garelick post, but I couldn't find his response to any website discussing the Oakley article. On the other hand, a commenter who is currently active on the Garelick blog is Tara Houle:

Tara Houle:

Great post Barry. I guess some are just jealous that Barbara Oakley has made such an immense impact with her column – again. Some have suggested it’s received more emails than other columns in recent memory; I wouldn’t be surprised by this. Many just do not want to acknowledge how poor our math instruction really is in our schools. Her previous article was also thought provoking, and powerful, and proves how wrong Boaler, Meyer et al. are with promoting edu fads in our schools 

http://nautil.us/issue/40/learning/how-i-rewired-my-brain-to-become-fluent-in-math-rp

Michael Salter lives in Australia and he wrote a recent column on tutoring which would further support Barbara’s comments as well 

https://pocketquintilian.wordpress.com/2018/08/24/why-tutoring/. 

It’s only a matter of time before the gurus will be forced to deal with the silent majority.

I have plenty to say about this Houle comment. First of all, to whom is she referring when she mentions "the silent majority"? I'm wondering about the majority of students who are actually sitting in the math classes. Some of the students might enjoy traditional math and are grateful for the p-sets because they help them learn math better. And other students don't like traditional math and wished that math their teachers would give them anything other than endless p-sets to do. Now take a wild guess which set makes up "the silent majority" of students actually sitting in math classes.

Houle mentions a "Meyer" as someone "promoting edu fads" in our schools" -- clearly, she's referring to former MTBoS King Dan Meyer here. But she also mentions a "Boaler" here. I notice that in that Number Talks book that I purchased over the weekend, the Foreword there is written by Jo Boaler, a Professor of Math Education at Stanford. So it's reasonable to assume that she is the "Boaler" to whom Houle refers here.

As I wrote yesterday, I still haven't decided to make Ruth Parker's Making Number Talks Matter my next side-along reading book yet. Still, let's look at Boaler's foreword as part of the current traditionalists post here. She begins:

"Number Talks are the perfect start to any math class. They are like no other teaching method I know."

Let's skip to the part that's most relevant to the traditionalists' debate. Boaler writes:

"In recent years, I was teaching a group of disaffected seventh- and eighth-grade students (Boaler 2015) with my graduate students at Stanford, in the challenging context of summer school. Most of the students hated math, and their only experience of math learning had been silently computing problems on worksheets -- hence their dislike of the subject."

Hmm -- here Boaler makes it sound as if not only do these middle school students hate math, but that they hate it because of all the p-sets. But according to Garelick, his own middle school students actually like it when he assigns them traditional p-sets. And according to Houle, a "silent majority" wants there to be more p-sets in math classes. Yet this surely doesn't sound like anywhere near a majority in this math class.

Indeed, these are probably students who, when assigned p-sets, are inclined not to do even problem #1 on the p-sets, since they hate doing them. That's why they find themselves stuck in summer school in the first place.

The traditionalists like to emphasize that their methods are more likely to lead to success at four-year universities -- including Boaler's own Stanford. To them, perhaps she should have asked those Stanford grad students what math they took in middle and high school to be admitted to Stanford. Of course, the traditionalists are expecting answers such as "Algebra I in eighth grade" and "AP Calculus senior year." And if their middle school teachers taught them anything other traditional math, the traditionalists expect "my parents sent me to Kumon" to be part of the grad students' responses.

Boaler continues:

"This experience was transformative for the students because they have never realized that math problems could be solved in different ways, particularly bare number problems such as 27 * 12."

To traditionalists, there is only one valid method to find 27 * 12 -- the standard algorithm. To them, any time spent on any method other than the standard algorithm is wasted time. To them, it's better for them to master the standard algorithm as quickly as possible so that the remaining time can be spent applying 27 * 12 to more complex problems (such as an area problem similar to the Pappas problem for today). That way, they can get a good grade in eighth grade Algebra I, a passing score on the AP Calc exam, and a fat admissions envelope from a university such as Boaler's own Stanford.

Later on, Boaler writes:

"One reader of my book What's Math Got to Do with It?, which shows several methods to solve the problem 18 * 5, said she had known that number problems have different methods but somehow had always thought that creative and flexible solutions were 'against the rules' in mathematics."

Well, they're definitely against the traditionalists' rules. I'm only two pages into Parker's book, and I can already tell that this is a book that traditionalists won't like. It's no surprise, then, that Houle lists Boaler among the educators whose teaching methods she disparages.

Boaler writes:

"When any teacher asks me, 'How do I develop number sense?' my answer is short: Number Talks."

When any teacher asks the traditionalists, "How do I develop number sense?" their answer is short: traditional math. In fact, the traditionalists would argue that Boaler harmed her summer middle school students. To them, her methods were blocking them from going on to earn degrees at Stanford.

Much of the traditionalists' debate revolves around textbooks. To the traditionalists, the new texts go too far in promoting reform methods over traditional math. Yet to many MTBoS teachers, the new texts don't go far enough. In order to help students learn, the texts must be supplemented with Number Talks and other activities that aren't in the text.

Actually, let me take that back. There's one curriculum that contains all the extra materials that reformers say help students learn. There's no need to add any extra activities, because the curriculum already supplies everything. And that curriculum is -- the Illinois State text that I used at my charter middle school two years ago.

I was thinking about what Sara VanDerWerf wrote -- how all math teachers in grades 6-12 (or is it grades 4-10?) should use Number Talks in their classes. But in yesterday's post, I wrote about how another Sara(h) -- Sarah Giek -- used manipulatives, not Number Talks, to teach exponents. Giek's class is more in line with the Illinois State text and its DIDAX manipulatives.

Indeed, we see that there really isn't much room for Number Talks with the Illinois State text. For according to Boaler, she used Number Talks during the first fifteen minutes of each day of her summer class. But Illinois State already prescribes a Daily Assessment for the first five minutes of each class. Instead, it's the DIDAX manipulatives that allow students to gain number sense -- just as Giek's manipulatives helped own her students.

If I ever teach a math class in the future, implementing Number Talks might be a good idea since there probably won't be the Illinois State text at that school. But at my old school, the Illinois State text, with all of its projects and manipulatives, rendered Number Talks redundant. All I had to do was implement the Illinois State text fully and my students could have received much of the number sense help they needed.

What purpose, then, would the MTBoS have served during my year of teaching? After all, for any project or activity that I read about on the MTBoS, there's an Illinois State project that I could and must give instead.

Well, Illinois State doesn't prescribe opening week activities. Many MTBoS activities, such as the VanDerWerf name tents, might have worked in my class. And while Illinois State prescribes the Warm-Ups, it doesn't say anything about Exit Passes. Even though the idea of making the date into a math problem is attributable to Pappas, a few MTBoS teachers have done the same. The Pappas daily problems fit as Exit Passes since Illinois State is silent here.

But most importantly of all, the MTBoS would have helped me gain confidence. When I first learned about the Illinois State text and its components, I didn't make the effort to learn how to use them because I didn't realize how helpful they could have been to my students. Seeing other teachers write about their own projects could have inspired me to work to implement the Illinois State projects.

And besides, Twitter is a two-way street. I could have posted pictures of my own Illinois State projects (and fulfilled today's Blaugust prompt), and other teachers might have been interested in how they could use those projects in their own classes.

I am not a traditionalist -- but on some days, I acted like one in class. Of course, for most teachers, traditional math is familiar -- and at the time I taught, the unfamiliar scared me. (Again, as I mentioned in previous posts, I do agree that some traditionalist ideas do have merit.)

Most modern texts allow progressive teachers to supplement their text with Number Talks and other progressive activities. They also allow traditionalist teachers like Garelick to supplement their text with Dolciani and other traditional lessons.

But with the Illinois State text, progressive teachers don't need to supplement the curriculum because it already contains so many progressive components. On the other hand, there's no room for traditionalist teachers to supplement the text with any traditional lessons (beyond what's provided in the "traditional textbook" component).

I'm glad that Garelick doesn't teach at a school that uses the Illinois State text, because he would have considered it to be torture. But many MTBoS teachers might enjoy the Illinois State activities. If I'd been a better MTBoS participant, I might have been more appreciative of the Illinois State text, since I was already handed activities, while other MTBoS teachers had to develop activities themselves (or find another MTBoS member who already developed the activity).

Today we'll check out the blog of another Blaugust participant, Benjamin Leis:

http://mymathclub.blogspot.com/2018/08/parabola-coordinates.html

Apparently, Benjamin Leis is not a math teacher. Instead, he appears to be a parent in charge of an extracurricular math club at his middle school.

It's actually difficult to classify Leis as traditionalist or reform-minded. In his math club, he teaches above-grade level math to all of the middle school students (in order to prepare the group for various math competitions). Traditionalists should appreciate the above post, where he provides formulas for the coordinates of the key points of a parabola. They like it when young students are allowed to learn Algebra I as early as possible, as opposed to being hampered by Common Core pacing.

On the other hand, in some of his earlier posts Leis mentions puzzles that resemble those I've seen elsewhere on the MTBoS, as well as some computer software. Traditionalists do not like the increasing reliance on technology. Then again, they might not mind puzzles and software in an extracurricular club as opposed to the main class.

I suppose it suffices to say that Leis defies classification into traditionalist or reformer -- just as Hung-Hsi Wu warns us to avoid false dichotomies. After all, I myself want to take ideas from both camps, but I admit that I lean more towards reform in the higher grades (and I had no choice but to support reform in my old charter class).

OK, let's finally post the Benchmark Tests. These are based on old finals posted to the blog. I admit that the tricky thing about Benchmark Tests in Geometry is that the students are coming off of a year of Algebra I, when they've thought little about Geometry at all. This is different from Benchmark Tests in middle school or Integrated Math, where there should be some continuity from year to year.