My Feelings about the SHSAT & Specialized Schools #iteachmath #MTBoS

I have a lot of feelings about the SHSAT, the exam in NYC used to screen students for the specialized high schools. I’ve taught 6th, 7th and 8th grade at three different middle schools so far (in my 8 years of teaching) – a new and small school in the Bronx, a large comprehensive middle school in District 2, and a selective, screened middle school in District 2. I’ve taught a variety of students in my classes, from a wide variety of backgrounds and prior math experience. I noticed several things that I don’t think gets talked about enough in this on-going debate about the Specialized High Schools, so I’m going to talk about some of those issues here.

The first thing I want to mention is that I’ve noticed is Saturday School. Many of my Chinese students told me they went to Saturday School since elementary school where they often learned how to speak Chinese and sometimes where they received additional math instruction. Depending on the teacher and the specific program, some of those students received very procedural-based instruction, but some of them did gain a better depth of understanding or mastery of the content. Sometimes, they were very familiar with “famous problems’ sophisticated solutions” like the formula for Gaussian addition or the locker problem, which meant they had a mental framework for recognizing when new problems were similar to something they had seen before. Assuming a 4 hour class every Saturday, for about 30 weeks per year, for the 8 years of K-7, that’s an additional 960 hours of math experience. Never mind the fact that some of these schools ALSO gave the students HW to complete and many students may ALSO go to class after school, during the week as well. So some of their improved performance in math class may be  due to gaining more experience with the math they had to do in school. (I have a whole post brewing where I’m convinced that much of what we test for with “giftedness” screening is ACTUALLY just “experience” screening).

The second thing I wanted to mention about the SHSAT is the content that’s covered on the exam actually necessitates test prep. Supposedly, the version given to 8th graders only includes math covered by 7th grade standards or “below,” but based on the types of things I’ve seen my students learn in SHSAT prep, I’m doubtful. Let’s take a look at some of the questions that might necessitate more study than just your “standard” 7th grade math curriculum. These questions below are all from the 2019 SHSAT handbook’s sample exam:

“62. The sum of two consecutive integers is -15. If 1 is added to the smaller integer and 2 is subtracted from the larger integer, what is the product of the two resulting integers?”

Theoretically, operations with integers is covered by 7th grade, however the word “consecutive” is not NECESSARILY covered (though it could be). So one part of this is about the vocabulary. If I’m a teacher who’s familiar with the kind of vocab on the SHSAT, I can be sure that I’m giving students access and exposure to hearing, seeing, and using that word in class repeatedly. If I’m a less experienced educator, then I might not know to give those kinds of problems to my students. Furthermore, if I am just problem solving this, without experience with other consecutive number problems, I’m more likely to take longer to figure it out, whereas if I’ve gone to test prep, I probably have a formula memorized for dealing with consecutive numbers or I have had enough experience that I know I can say the consecutive numbers are x and x + 1 and I can write an equation to solve. x + x + 1 = -15 so 2x + 1 = -15 so 2x = -16 so x = -8, and therefore x + 1 = -7. Then, I can change them so that 1 is added to -8 becoming -7 and 2 is subtracted from -7 so it becomes -9. The product therefore is +72, which you would then have to grid in.

There’s a problem requiring students to have memorized geometric formulas (including for surface area of 3D shapes like prisms and pyramids), another one requiring students to understand how to graph a compound inequality (which is NOT technically included in 6th or 7th grade, though some teachers might teach it).

“97. In the set of consecutive integers from 12 to 30, inclusive, there are four integers that are multiples of both 2 and 3. How many integers in this set are multiples of neither 2 nor 3?”

Once again the phrase consecutive integers is used, but now an additional vocabulary word, “Inclusive” is used as well! Neither are necessarily part of the MS curriculum by default.

“91. There are 6 different cookies on a plate. Aiden will choose 2 of these cookies to pack in his lunch. How many different pairs of 2 cookies can he choose from the 6?”

This problem is technically a combinations problem, and technically, formal combinations aren’t taught until high school. An MS student who’s never seen a problem like this can probably work it out (especially with numbers this small), but a student who has memorized a formula can work it out more quickly. Factorials aren’t even technically part of the math curriculum up until this point, so while students who’ve done the Four 4’s activity might be familiar with them, it’s not something that’s “safe” to assume. By the combinations formula, 6 C 2, you would do 6! / 2(6-2)! which is (6 * 5 * 4!) / 2(4!) OR (6*5)/2 or 30/2 which is 15. With a number such as 15, you could also make an ordered list, if you called the cookies A, B, C, D, E and F, and then showed AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF, also getting an answer of 15. But if you weren’t familiar with this type of problem (which is not precisely required by the math curriculum up until 7th grade), you might not know about it. Plus, the counting principal (which helps you “discover” factorials) is also not a necessity. Again, a student who’s been exposed to problems like this one can easily solve it, whereas a student who has only ever had “school math” (especially if taught by a teacher who DOESN’T know what’s on the SHSAT), might need to spend more time “problem solving” what appears to be a novel problem instead of applying a familiar strategy to an exercise they recognize.

“92. For a presentation, Deion can create 5 slides in 20 minutes, working at a constant rate. Kyra can create 3 slides in 10 minutes, working at her own constant rate. What is the total number of slides the two of them can create in one hour?”

Now, again, problems like this technically are within the scope of the 7th grade curriculum, but once again, this is a “type of problem” that is super familiar if you know this structure, but very complicated and prone to misconceptions if it is novel.

“99. A box contains 5 strawberry candies, 3 banana candies, and 2 orange candies. If Braden selects 2 candies at random from this box, without replacement, what is the probability that both candies are not banana?”

When I taught 7th graders probability, we only covered “with replacement.” WHOOPS! I’m not 100% clear that without replacement (conditional probability) IS in fact a 7th grade standard.

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“105. In the infinitely repeating decimal above, 7 is the first digit in the repeating pattern. What is the 391st digit?”

Although this problem would APPEAR to be covered by this standard: “CCSS.MATH.CONTENT.7.NS.A.2.D Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats,” solving this problem efficiently actually requires using modular arithmetic (or at least thinking cyclically). For example, if I recognize that there are 6 digits in the repetend of the 13ths, then I know that to get out to the 391st digit, I need to repeatedly write those same 6 digits. Therefore, I can do 391 / 6 to get 65 and 1/6 or 65 remainder 1. That tells you that the 391st digit is the same as the 1st digit, which is 7. If I’m familiar with this style of problem, then it’s a relatively straight-forward division and remainder problem. If I’m only familiar with the actual standard, then this problem might induce me to tears, thinking about having to write 391 digits to find out what number is in that place!

“86. If w – 1 is an odd integer, which one of the following must be an even integer?”

A) w + 1, B) 2w + 1,  C) 2w – 1, D) 2w – 2

While writing algebraic expressions is technically part of the 7th grade curriculum, if you have increased familiarity with odd and even numbers being represented algebraically, then you can recognize that if w – 1 is odd, the next odd integer will be w + 1, whereas w itself must be even (due to the odd/even alternation). Thus, A can be eliminated. Looking at the remaining options, if w is even, then 2w must ALSO be even, so D must be correct, as that takes an even number and subtracts 2 getting to a smaller even number (whereas subtracting or adding one from an even number will give you an odd number). I don’t “mind” this question as much as many of the others, but I do think it requires a level of sophistication with algebraic notation that is NOT normally expected from the 7th grade math content standards and may or may not be gained from your typical math classroom instruction, as dealing with algebraic notation often doesn’t begin until middle school for most students (SOME get it in 5th grade, but not before), whereas dealing with even and odd numbers often happens in elementary school – and rarely to the degree of generalization required to solve this problem.

There’s also some geometry vocabulary that’s not normally used in MS:

“92. R, S, and T are midpoints of the sides of square MNPQ, as shown above. What is the sum of the areas of the shaded triangles?”

While midpoints aren’t unreasonable to infer the meaning of, they’re not typically talked about much in middle school – unless of course you have a teacher who knows to cover them! Midpoints aren’t officially covered until the high school geometry standards!

From 2017-2018:

“74. A large circular dinner plate has a radius of 20 centimeters. A smaller circular plate with a circumference of  centimeters is placed in the center of the larger dinner plate. What is the area of the part of the larger dinner plate that is not covered by the smaller plate?”

To solve this problem, you must know the relationships between circumference and area of circles (and their diameters/radii) without having those given to you. Dealing with pi also often adds an extra challenge for students, though that doesn’t necessarily rule this question out as doable.

In 2016-2017 or 2017-2018, the SHSAT got updated to better reflect the 7th grade standards (though I confess I haven’t actually seen a REAL SHSAT, because those are sealed and not released by the city/state). Prior to then, there were more questions on the test that required additional instruction (also, it included more 8th grade topics, but since it’s given in October of 8th grade, depending on your school’s sequence, you might not have covered those topics by then!). There were a bunch of questions about prime factorization (not officially a standard, though it often gets covered in 6th grade or before) and several involving scientific notation (which was often covered in later 8th grade).

Actually, now that I’ve looked over the SHSAT materials in more detail, I’m actually wondering why so many of my students were taught the Pythagorean Theorem by the test prep classes, as well as operations with radicals – those DON’T seem to be covered on the exams (anymore – I only went back to 2015-2016 for this, so maybe they used to be?).

Also, as an aside, I wonder what the questions themselves are actually screening for. Here’s one that I’m not convinced will tell you who’s going to excel at the specialized high schools:

What is the greatest common factor of 2,205 and 3,675?

A. 147 B. 245 C. 441 D. 735 E. 1,225

To be calculated without a calculator! Now, thinking about the steps you would need to take for this one, you have a few options: you can just try dividing the two numbers by each of the 5 options… But the long division by hand is bound to be a pain in the ass. Alternatively, you could try using the “ladder method” or prime factorization to find the GCF of 2,205 and 3,675… but starting with 3 and 5, you might not easily get to 735 – certainly not quickly! So there’s a potential waste of time.

What is the greatest prime factor of 5,355? F. 17 G. 51 H. 119 J. 131 K. 153

Another one where I’m not convinced it’s a good use of time. 51 is not a prime, neither is 119, nor 153. For 51 and 153, they can “easily” be identified as divisible by 3 (IF you know the divisibility rules – which I don’t think are required by the standards) by adding up the digits to get 6 and 9, both of which are divisible by 3. 119 is not divisible by 3, so it might take a little bit more work to verify that it is not prime: If you happen to know the rule of divisibility by 7, you might recognize it, or you could just try dividing it by 7 (personally, that’s my go-to, as I can’t remember the divisibility rule for 7s). Still, you’ve wasted a bit of time ruling out the non-primes, and you still have 17 and 131 to check. I suppose you can start with 131, as that is the bigger of the two numbers. Dividing 5355 by 131 will take some time, but you can see that it will give you 40 with a remainder of 115. So by process of elimination, it must be 17, and you can skip the extra division to confirm that.

Now, I’m going to go back to my original claim: I don’t think the standard mathematics curriculum IN MIDDLE SCHOOL CLASSES will prepare a student to successfully get into one of the specialized schools without going to extra math classes (and ELA, probably, but I’m focusing more specifically on the math, because that’s my area of expertise). Now, when I consider the ways in which teachers are prepared to teach and focused on their curriculum development, I would say that most middle school teachers (unless they’re teaching an SHSAT prep course) have never done an in-depth analysis of the kinds of questions on there, and are not looking to embed those questions into their curriculum or deploy them as extension/challenge questions for students who finish early (even though there are actually some questions that could be tackled by kids as young as 6th grade! And I may steal some this year!). So there’s an opportunity gap here: knowing what the exam is and recognizing that you do actually need to prepare for it. The DOE’s website is misleading about it (bordering on lying, in my opinion): “The test measures knowledge and skills students have gained over the course of their education. Keeping up with schoolwork throughout the year is the best possible preparation.” I don’t disagree that you can’t do well on the SHSAT if you haven’t done these things, but I honestly don’t think that’s sufficient. And to tell the students that it is sufficient is to rob them of the preparation time.

Now, some of the test-prep classes are very procedural and focus on memorization of formulas and “drill and kill” on a variety of the “most common” types of problems, showering kids with tons of worksheets and packets to practice skills they are shown in classes on the weekend or after school. Sometimes, these courses include content beyond grade level and may “ruin” discovery-based lessons (as was the case at my last middle school, where most of my 7th graders ALREADY KNEW the Pythagorean Theorem – which is technically an 8th grade standard!!).

The second big issue that I have with the SHSAT (and testing in general, to be honest), is the amount of money we pour into it that COULD be going to bettering school programs. Consider for a moment the fact that the testing industry is a $2.5 billion industry in the US (in 2015), and yet we can’t find the funds to fix the heating system in Baltimore schools, the water system in Flint, or any of the host of other problems that affect student learning(often in poverty)? New York City is currently paying Questar $44 million for 3-8 ELA and Math state exams (when they won out the current contract over Pearson), and according to the NYC Data website, our current budget spent $458,923 in FY 2015 and $542,515 in FY 2016 on the SHSAT alone (and another $100,000+ on the gifted and talented screening). Consider what we could do for our students with an extra half million dollars! And that doesn’t even include the money being spent on offering “free” tutoring – it’s free to the students, obviously, but not free for the city to provide – that’s gotta be budgeted for somewhere (though I couldn’t find the details on that cost, as it’s broken up into too many different programs that I don’t know all of the names for – there’s the DREAM program, among others). Plus, there are private companies that do test prep as well – some free or application-based while others for profit. All of the money that’s flowing in those pockets could be going towards improving ALL of our schools!

There are a variety of articles about specialized HS admission in general, the $13.4 million that the city pays Pearson over 6 years for administering/creating (scoring?) a test that no one knows the validity of, the current debate over eliminating the SHSAT, the reaction of Stuyvesant Students to the proposed changes, the culture of test prep, the monetary bonus that the specialized high schools get (which is part of why they can offer so many extras – that and their size in general!), the problems with (un)fair funding of schools in general, the new Chancellor’s reaction to screening/admissions policies, but I’m going to focus on some of the issues I think are missing from these debates. Or at least, the ideas that have occurred to me.

Now, let’s put the SHSAT aside for a moment, and handle my other concerns about the specialized high schools in general. Something that I don’t think is talked about enough is that the culture of those schools is very competitive, and not every student thrives in that kind of environment. I know for myself, when my parents asked my 8th grade self if I wanted to apply to the specialized high schools (and leave Catholic school), I declined. At the time, I said that I would rather be a big fish in a little pond than a little fish in a big pond (but not every 13-year old knows that about themselves!). I recognized even then that a school like Stuyvesant takes the students who were at the top of their class in middle school, and then immediately, half of them make up the bottom 50% of the class! I knew that level of competitiveness would be demoralizing for me, and I chose a more supportive school environment – and I’m glad that I did.

So this leads me to one of my big questions: what is it about these schools that makes them “specialized”? Supposedly, some of it is the culture of the students who “worked so hard to get in” that they continue to push each other forward – but I’ve never actually attended their classes, so I can’t really comment on it (though I did observe at Stuy when I was in grad school, I remember being bored by the lectures and wondering if the school catered to kids who thrived in that particular kind of classroom set-up). I know they’re able to offer more higher level math classes – but to some extent, that’s because they expect all 9th graders to come in having completed AT LEAST algebra 1 (which is typically 9th grade math in NYS, but many 8th grades offer regents courses to “accelerate” students). To some degree, because of the size of the schools (and having enough money to pay veteran teachers), they’re able to offer a variety of courses to students – things that are more specialized or specific than the general offerings of most NYC high schools – but that’s about access to resources. I know some of the educators who work at the various specialized high schools (mostly through MfA), and I don’t know that there’s anything “magical” about their curriculum choices or their pedagogy that other good teachers can’t also implement at their own schools. But, I do know that many of the teachers at the specialized high schools have MANY YEARS of teaching experience under their belt. And while it’s true that years alone is insufficient to guarantee a good teacher, the fact that so many of them are MfA Master Teachers speaks highly of their pedagogy. And this brings me to an article about segregation of the NYC schools in general that I was reading recently. When it comes to school segregation, “New York City is among the worst offenders. Among the city’s 1.1 million public school students — the largest school system in the nation — children of color have an 80 percent chance of attending a school where the student body consists of fewer than 10 percent white children. Fifty percent of white students attending New York City public schools are concentrated in 7 percent of the schools” (from the article I just linked to).

Continuing to look at that article sheds some light on the funding inadequacies that set some schools up to have skilled practitioners in front of every student with money spent on professional development to enhance their teaching quality, whereas schools in neighborhoods with children of color and people of lower economic backgrounds struggle to recruit and retain quality teachers. Many teachers at those struggling schools are within their first five years of teaching – I taught at a school that was struggling in the South Bronx my first year teaching, and among our teaching staff, there was an incredibly high number of first year teachers (something like 13!?). A student could go through 12 years of education and have a first-year teacher EVERY YEAR. While it’s true that some first year teachers manage to overcome the obstacles to do good work, I think even exceptional first year teachers become better with experience, and their later students benefit even more from their skills. Additionally, PA (parents’ associations) at schools raise very different amounts of money to support the school, depending on the financial capacity of the parents – at both of my last two schools, the PA’s managed to raise thousands of dollars to pay for arts instruction and partnerships beyond what the parents at my first school could ever have afforded.

With all of our “school choice” initiatives in NYC, we have created schools that are “dumping grounds” for the students that “no one else wants” – for example, in District 13 (Unison’s School’s district), 25 of the 27 publicly funded schools (including charter schools) require participation in an application process. “But as I see it, what the city described as competition turned into a segregation filter: Choice was only an option for those with the time, literacy, and determination to navigate a complex and nonstandardized admissions system. These are major hurdles for the most vulnerable families. The burden of their failure to navigate such a system hurt no one more than their children, who no rational person would argue should be able to manipulate this complexity at the age of 11.” So in effect, school choice has reinstated and reinforced school segregation in NYC. With charter schools expelling (sorry, counseling out) students twice a year (in October, after they get the money for kids with disabilities and in April, right before the state exams), there’s an influx to the schools with open seats – which are often the schools few people who understand how to “work the system” would ever choose.

Now, you might have noticed that I’ve zoomed out from just focusing in on high schools to looking at the quality of schools across the grades, from PK (and 3K!) to 12th. A big part of that is because you can’t just start trying to “equalize” the playing field in high school – it needs to start in pre-school. Otherwise, you might be setting up a student for failure. Going back to my own experiences, when I got to engineering college, I struggled in my math classes because I’d never developed the types of math study skills I needed to succeed, as I’d always been a “good student.” At the time, I had a fixed mindset, and I thought my struggles meant that I had reached the limits of my capabilities and that I didn’t belong there (especially when compared with my peers who seemed to get it easily). I now recognize that if I had taken the time to form study groups, go to tutoring (not just office hours), and to try to make sense of the material conceptually (even though it was more often taught procedurally to me), I might have done better (though no guarantee that I would have enjoyed it any more!). I also couldn’t easily develop these skills in college, having never needed them before.

So this brings me to the next part of this debate. This interesting article talks about expanding access to the gifted and talent program in elementary schools (I would bet some districts, more heavily populated by Black and Latinx people, have fewer GT programs in general) and maybe even creating a GT program for middle schools (since right now, it ends in 5th grade, but then the specialized HSs are essentially thought of as the GT of HS, so there’s a weird 3-year gap where some middle schools have either tracks or just general acceleration that take the place of GT unofficially). Now, this article also raises some points about why this increase in access to GT programs wouldn’t actually address the underlying segregation issues, two of which I’ve copied here:

  • “Gifted programs are already deeply segregated, so an expansion isn’t likely to spur more diversity absent other measures to increase integration.
  • Gifted programs would still rely on an entrance exam to determine admissions — and create a new test for middle school students. Integration advocates trace the diversity problems in gifted — and specialized high schools — back to admissions tests. Critics say that the tests advantage students with the time and resources to prepare.”

My last school was one of those middle schools that unofficially thought of itself as a GT school – it was certainly a feeder school for the specialized high schools, as many, many, many of our students went to them. And this leads me to the other reason you can’t start trying to equalize the playing field in HS – if the ability to solve math problems well is partially related to experience solving math problems (i.e. when you recognize that a new problem looks familiar, you’re more easily able to find a strategy to solve it), then more experience solving more complex math problems will result in a student who is more prepared to solve other more complex math problems. What one 8th grader knows at one school in NYC is NOT uniform across the board. My last two schools showed me that. I was teaching my 7th graders at the screened school things I didn’t even teach my 8th graders in the non-honors program at my prior school! We taught them about the Pythagorean Theorem in 7th grade (instead of 8th) and we included instruction about the “special” right triangles (30-60-90 and 45-45-90), and solving problems involving composite figures, inscribed figures, and circumscribed figures. And if they were in the accelerated class at my last school? Well, they even learned about trig ratios! Now, while none of that is officially on the SHSAT, if you think about the skills needed to solve those problems (comfort drawing in lines that aren’t there, comfort reasoning about diagrams’ angles and side lengths, etc.), they’re certainly honing that skill, and more comfortable deploying it on a problem on the SHSAT.

Now, this brings me to a small aside that I need to talk about. And I say this as someone who was in a GT program in elementary school (both for the 1st and 2nd grade that I was in public school, and the enrichment program in 3 – 8th grade at Catholic school). I’m not 100% convinced that “giftedness” exists or can be tested for or that it MATTERS. I have begun to think that much of what we think of as “giftedness” is actually related to a combination of experience and deliberate practice (though the 10,000 hour rule popularized by Malcolm Gladwell has been “debunked,” I still think there’s something to it). I’m really not convinced that there IS such a thing as “giftedness” that requires a DIFFERENT school nor that “giftedness” is actually a meaningful predictor of… anything. And I’m not alone! There’s research and articles debating whether or not giftedness exists and whether it matters. And our GT programs screening tests are norm-referenced, which means the students get scored against each other and then are placed based on being in the top ten percent… Or top three percent, depending on the space in the program! There is much controversy surrounding the history of IQ testing and race and eugenics, which I don’t think is talked about often enough by the people who tout IQ tests and giftedness tests as somehow being able to see the potential inside someone and give them more resources so they can reach their full potential (rather than recognizing that IQ testing began as a way for us to judge some humans as “more worthy” than others). I think there’s a fundamental, unanswered question here, which is “Is every human child in our school system equally worthy of our time, money, and resources?” And, sadly, the answer is often “No, some kids deserve more,” though the actual measure we use to determine WHICH kids deserve more varies person to person (and I should say, as a person committed to equity and justice for all, I acknowledge that some kids need MORE of the system’s money to support them in overcoming obstacles that other people don’t have to face, so please don’t misinterpret my statement to mean that I think we should spend the same amount of money on each kid – in fact, I think the kids who are struggling the most probably need more of the right resources dedicated to supporting them in achieving their dreams and being successful).

Now, I read articles like this, that insist on giftedness, and I think it’s important to specify here that I DO believe in neural diversity. I have recently discovered that I have ADHD, and my stepdaughter is on the Autism spectrum. Both of these things mean that our brains work differently than neuro-typical brains. This means we might react differently to stimulus than other people. This means that we might benefit from different types of activities or instruction than others – but it doesn’t mean we should be isolated in a room with only people who have the “same kind” of  brain we do (I put same kind in quotes because I don’t even think that another person with ADHD is necessarily “the same kind” of person I am – there are too many factors and variables for that one feature to necessarily link us).

I’m going to share an anecdote with you from my own personal life experience. My mom is a professional reader – no literally! Before retiring, she was a book editor. She has her PhD in literature from Harvard. So she has a lot of experience reading and a LOVE of reading. When I was a baby, she read TONS of books to me. I started pre-literacy stages as young as 2 (I was turning pages and reciting the words to my favorite book, “Chicken Soup with Rice by 2), and by kindergarten, I read over 200 books as part of a school “read-a-thon.” In first grade, I read the Lion, the Witch, and the Wardrobe, independently and had no idea it was “technically” a fifth grade reading level. I continued to read well above grade level throughout my life (though I often chose/choose to read what my middle school ELA teacher, Mrs. Smith, called Brain Candy, such as romance novels from my mom’s publishing company or genre fiction, which I still think is highly intellectual, but often scoffed at by literary folks). So let’s examine this situation: what’s the cause of my above-grade level reading? Is it the habit of going to the library every week as a child that was instilled by my mom? Is it her genetics of “good reader” that got passed on to me? Is it the sheer quantity of words and books I had read by the time I was 2? I’m not sure – but I’m also not sure that it matters.

I think that what’s MORE important is the way in which we (teachers) react to students who are presenting like I did. When I was in first grade, I went to my local public elementary school. I remember we used to use these readers where we had to read a short passage and then answer some multiple choice questions about it (presumably to assess our comprehension). I don’t recall anything specific about them beyond that, though I do think they seemed ancient and old fashioned to me at the time. I remember getting bored because I would devour those readers, and be done with five of them in the time that many of my classmates needed to read one or two of them. So I would often start talking, to engage with other students, to be stimulated again, and to enjoy myself. My mom tells me that my first grade teacher (a second year teacher) moved my desk into a corner by myself because I was too chatty (until my mom came to the school and protested, using her white, middle class privilege to demand that I not be in trouble because of my speed at reading). Now that teacher was feeling overwhelmed because she didn’t know how to handle me (and the multitude of other challenging behaviors the 30 first graders were giving her, especially when the previous year, she’d only had 25 kids). I sympathize much more with her now, having been in her shoes as a novice teacher. (I should also point out that I was in my school’s GT program at the time, but it was a small pull-out session, once a day? a few times a week? I don’t recall, so I was mostly in the gen ed room for most of the day). I only lasted in that public elementary school for two years before my mom decided to move me to Catholic school (where I flourished – but that’s a post for a different time).

However, what I think was lacking there, and often lacking from the discussion of “gifted education” in general is differentiation. I honestly think that if my teacher had been skilled in differentiated instruction (and probably had a smaller classroom size and more adults in the room!), the situation might have been very different. I think differentiation done right is one of the TRICKIEST things to do and yet, also, one of the most important things. I think there is intense value in NOT segregating kids by “ability” – I think we need to value each other, and the community that we can build when there is true diversity in the room. I won’t lie and pretend like I’ve got it all figured out: I’ve been teaching kids 8 years now, and I still struggle with differentiation at times – how to plan it, how to enact it, how to keep it manageable. I’ve done work around improving my differentiation with Rhonda Bondie, and I definitely think she’s on to something there. I also think that rich problems provide ample opportunity for self-differentiation (my preference, as a teacher!), where students can choose what avenue to explore, can deepen their understanding at its particular edges, etc. I think being given the right environment, where you can explore and ask your own questions – and try to answer them (even in math class!) is far more valuable than being mindlessly zoomed ahead through a “standardized curriculum,”

The reason I bring up all of these issues in a discussion about the specialized high schools is because I want to get to the heart of the matter. This question about how to ensure black and latinx students are equitably represented at the specialized high schools is really a different question: how do we ensure that ALL students in the ENTIRE city, no matter their zip code, etc., have a high-quality education that prepares them to be a successful adult? And that’s a harder question with no easy answers.

ALL students (not just the 18,000 served by the specialized HSs) deserve a quality education that prepares them for adulthood. ALL schools should be able to provide a quality education for ALL students so that school choice becomes an irrelevant detail. Instead of focusing on the “quality” of the schools, school choice (if we continue that system) could allow students and families to focus on other aspects of a school’s culture – perhaps the special focus (i.e. choosing to go to the academy of film and TV because of an interest in becoming a camera operator) or choosing a school that is known for trips to museums or whatever other features of a school make it unique – but ALL students are assured that the quality of their education would be equitable, no matter the school they select. To do that, I fully believe we need to tackle integration (which we’ve been discussing for YEARS, probably even decades, as this article about how to tackle segregation is from 2015), equitable school funding, teacher recruitment, retention, and training, especially focused on practical differentiation, and incorporating WAY more culturally responsive/trauma responsive teacher training (and expanding restorative justice and the focus on community building within the schools). Honestly, if students from EVERY NYC HS felt confident that the quality of their education was as good as if they had gone to the specialized high schools, then there wouldn’t necessarily be as much demand over the 18,000 spots (in a school system that educates 1.1 million, that’s a tiny sliver).


Day 1 and Week 1

So I’ve been doing a lot of thinking and reflecting about the first two weeks of school for this upcoming year. I’m not sure yet exactly what I’m going to do now that I’m changing schools. I originally had my plan for the first two weeks (since there’s only 6 days of instruction) when I knew my school, but now that I’m going to be at a different school, and I’m looking at the prior year’s opening weeks, I’m not sure what to do.

Let me lay out what my original plan was and then I’ll think about which parts to keep and which parts to change.

I used to have my students fill out a questionnaire on day 1, but after a few years of never doing anything with that information, I realized my strength was NOT in reading those papers (and making meaning or being able to USE that information in anyway), so two years ago, I did away with the questionnaire. Instead, I replaced it with the name tents from Sara Van Der Werf as both the opener and closer on day 1 (and the whole first week of school we use it to communicate). I learned WAY more from that than from the long questionnaire, and because it was so short, I was able to respond to each and every student I taught, and it built a much more meaningful first day conversation. So I definitely plan to continue to use that in my new school.

The main activity that I’ve used on the very first day of school has been basically the same exact activity since 2012, my third year teaching. I did some googling and reading on the internet and I found two websites that talked about an activity “Numbers about Me” that the two teachers used almost as a “quiz” to get to know them. They also talked about the quality of responses the students told them about important information about them. And I will confess – some years, I’ve learned more important stuff than others (like “7, the grade I was when I stopped cutting myself” – though that student later cut again in 8th grade). I like it in many ways because it gives me an excuse/fun way to introduce myself to the students using numbers and I like it because it’s original – none of my students have done a similar activity in their other classes, and activities where they write the rules all wind up feeling the same after a while (especially when you ultimately have the same rules every year!). I have shared with them the following facts every year (in multiple choice quiz format and then there’s a “reveal”). I ask a question along the lines of “The number 1 is important to Mr. Golan because…” and then three reasons, two of which are usually humorous and one of which is true.

    1. Number of children Mr. Golan has.
    2. Number of dogs Mr. Golan has.
    3. Number of states that Mr. Golan has lived in.

I then share my answer with the kids (in this case, it’s the number of dogs I have).

Screenshot 2018-07-23 13.28.38.png

I go through that process of asking a multiple choice question and then revealing information about myself about 4 more times. Last year, I shared information such as the following with my students:

  • 36: Number of weeks Mr. Golan attended Space Camp.
  • 2: Number of siblings Mr. Golan has.
  • 109: Number of board games Mr. Golan owns.
  • 5: Number of dollars Mr. Golan spent to buy his bike.

I then ask the students to write about 3 numbers that are important to them and why. Most of my sixth graders wind up only getting through one or two numbers if I make them do it in class, in about 5 – 7 minutes of writing time. And then I always inevitably have the problem where some kids write faster than others, so one kid is done WAY early and some kids still haven’t even finished writing about their first idea. Some kids choose to make it multiple choice and others choose to tell me the narrative about why that number is important. Both ways tell me a lot about the kid in some ways, but I also don’t have a great way of tracking this information, so with my memory, I wind up forgetting it as I get to know the kids. I have the kids use this activity to do their first turn-and-talk, where they partner up and share what they’ve written with their elbow partner. It gives me an opportunity to introduce some norms for the first day of school about how we talk in partnerships.

I feel like there are many pros and cons to this first day activity, and I’m on the fence about whether or not to continue using it as DAY ONE. I enjoy being able to share about myself with the kids, and I think it builds nicely into the name tents activity at the end where I invite students to share something with me. I’ve debated whether to have students finish it for HW or not to add more to it. I’ve even had students choose one to make a “poster” of in the past, but other than putting it up on a bulletin board, I rarely wind up using that. I feel like it’s not something I get super engaged with learning about the students from, even though that’s my ultimate goal. I think it’s like that classic fire hydrant in the face – I am getting too much information to take in. The last two years, I’ve had students complete the “Who am I” handout from Dan Meyer, and I’ve retained that handout much into the school year. I feel like that information I’m much more likely to go back to, though there are a few that I want to change. Graduating class always confuses my sixth graders (and graduating from which grade? 8th? 12th?). I also feel like there were things I wanted to know about students that weren’t included in there, but I can’t recall them now, off the top of my head.

Recently, I was reading twitter and I came across a new blog from Jess. I really like what she described as her ideas for the first day of her classroom in her blog, but I’m not sure whether it makes as much sense coming from their math teacher in middle school… So I might also see what the other teachers are planning for their first day activities.

The rest of my day 1 tends to be collecting summer assignments and giving out the HW and supplies list/welcome letter and then giving students enough time to respond to the name tent activity. Although, that’s been in a 44 minute period in the past, and this year, the new school I teach at has hour long periods, so I might be able to do a little bit more on day 1… The biggest downside to day 1 as I see is it that it’s very me-focused in terms of teacher talk (the kids do a turn and talk about their important numbers, and I’ve typically asked a few kids to share out at the end – often about something they’ve learned about their PARTNER, as opposed to sharing their OWN ideas, but I haven’t done that in a few years). Otherwise, they really don’t have an opportunity to talk; they just do a LOT of listening! And, there’s also no math on day 1! So these are the two reasons I’m questioning my choices – while it IS a fun activity and helps build relationships (the number one priority in all of September), I worry that it sends the wrong message for the first day of class.

Anyway, we’ll revisit day 1 ideas after I discuss the rest of the first two weeks.

On Day 2, I’ve typically done a Growth Mindset introduction the last few years. The kids read over their name tent feedback and set them up, they watch some videos from Jo Boaler, and I have them do an exit ticket about “I used to think… but now I know…” regarding ideas around intelligence. This, combined with the posters that say “change your words, change your mindset” make for a great beginning of school year bulletin board. With this activity, I do a lot of turn and talks after each video we watch, and I have whole class share outs to discuss some of the ideas we’re hearing that may feel new or different. I really find this to be a valuable activity for the students, though I’m also wondering about combining it with the Talking Points activity that James Cleaveland created and shared here. I think considering my new school uses mastery-based grading (and Jump Rope) and my 6th graders may never have been assessed like that (and I’ve never done assessing like that), and it fits in line with some of the ideas about the way our intelligence can be grown (and not comparing ourselves with each other, but only with our past selves, etc), I think it might make a nice conclusion to day 2.

However, once again, day 2 concludes with no REAL MATH. I’m on the fence about this delay. On the one hand, I value the importance of community building and norm building and relationship building. I think that we can’t do any real math, I can’t ask my students to be vulnerable and trust in each other and me if we haven’t done the pre-work to set up that type of community… At the same time, how many days without math do we need for that? Is there a way to accomplish some of those same ideas WITH math embedded? I don’t have an answer for that.

Day 3 is another homage to Sara Van Der Werf, as I use her 100 numbers activity.  I found that it’s a great way to take photos of my students and discuss what good group work should look like and sound like. I think it’s especially important because one of my personal focuses this year is on the tension and balance between independence and interdependence (see my previous post), and I realized that sixth graders don’t always know how to listen to each other and do successful turn-and-talks! They don’t know how to share the air (or that it’s important that we SHOULD!), so I think it’s important to spend some time discussing these norms. I just noticed that Sara also includes print outs of the photos she takes, and I LOVE that idea – and I think I will try to use it this year! Once again, though, we have day 3 with minimal math. At least this day, there’s MUCH LESS of me talking, and way more of them talking. I also use this activity to discuss what mathematics IS (I think I had my students do the tweet “#mathis” activity last year  that I got from Sarah Carter’s blog, Math = Love).

Over the course of these three days, I elicit from my that math is the study of patterns and that mathematicians make sense of math by convincing themselves (through independent think time), convince mathematical friends, and convince skeptics. We use that word skeptics A LOT over the course of the year, and I have two posters to match those two ideas. In the past, I’ve used the next week of classes to further set up our problem solving and listening community by using math tasks from Jo Boaler’s Week of Inspirational Math. This year, my original plan was to use three contemplate-then-calculate tasks around area because my previous school’s first unit was going to be a modified Illustrative Mathematics first unit, which made area its first unit. My original plan was to do three days of CthenC, starting with the same pattern David Wees and I created to launch it this last year (basically the number of squares arranged in a rectangle, where the rectangle’s area is 2n + 2), then using the circles set up in an array with a “hole” in them from Illustrative Mathematics’ grade 6, unit 1.6.1, and then using the visual patterns #43. In thinking about each of those three tasks, I felt like they each leaned a bit more towards one of the three different types of “structural” thinking – the first one feels CONNECTED to area, the second one feels like they’re most likely to SUBTRACT the missing CHUNK (or possibly to rearrange), and the final one seems most likely to be CHANGED. That would give us some structural language to use from the get-go.

The BIG difference is that now, instead of my first unit being about area (and thus directly using those skills I was going to introduce from CthenC), my new school is using the CMP3 curriculum still. Our first unit is Prime Time, which deals with factors, multiples, LCM/GCF, prime factorization, the locker problem, even and odd numbers through rectangles, and eventually order of operations and the distributive property. I haven’t done the math yet for the distributive property problems in there, but I wonder whether it is introduced better there or in the IM curriculum – I’m going to decide over the summer which way to use to teach it. I also REALLY like the way I’ve done OofO the last two years through the NCTM article about “The Truth about PEMDAS,” but I don’t know if my students will need more of an introduction than the Boss Triangle, or if we can start from there.

I guess my big question is how to incorporate more math-doing to the first three days of school to get my students thinking like mathematicians sooner – but then also wondering whether I need to rush that, or if it’s ok to delay that for a few days… What are your thoughts? Feel free to reply here or to tweet at me @MrKitMath

Goal Setting in Math (part 2)

Tonight, I was talking with Rhonda Bondie about Goal Setting in math classes. I was sharing with her my observations from the previous post that most student goals either focused on their grades, their HW completion, or their participation in class discussions. It rarely focused on their actual math “skills” or strategies or thinking. I acknowledged that it was harder in math class because there was a tension in thinking classrooms between revealing too much information by revealing the topic of study too early (i.e. telling students that we’re going to discover the Pythagorean Theorem might ruin the discovery if a student has been introduced to the formula already in SHSAT prep or Saturday School, whereas telling students we’re looking for a pattern in the areas of the squares doesn’t give it away, but might make it a bit harder to set a goal).

Except, maybe it wouldn’t make it harder! Maybe THAT was the key to setting the goals. I realized that vague “we’re looking for a pattern” connected to what it is mathematicians do (look for, study and analyze patterns), and that maybe the Math Practices were a better way to set up our student goals. And then I made a connection to my absolute favorite diagram from from Amy Lucenta’s and Grace Kelemanik’s book, Routines for Reasoning.


In the book, they argue that the 8 Math Practices are not actually created equal, but instead, there is a hierarchy.

“MP 1: Make Sense of Problems and Persevere in Solving them is an overarching goal.” MP2: reason abstractly and quantitatively, MP7: look for and make use of structure, and MP8: look for and express regularity in repeated reasoning “describe three avenues for mathematical thinking” that allow you to solve problems. The remaining practices, MP3: construct viable arguments and critique the reasoning of others, MP4: model with mathematics, MP5: Use appropriate tools strategically, and MP6: attend to precision, all describe important ways to navigate those three avenues of thinking and play an important role in problem solving, no matter which avenue of thinking you choose. (I’ve paraphrased/summarized/quoted various ideas from pages 3 – 10 in their book in this paragraph)

This made me think about introducing the following ideas to my students at the beginning of the year.

“In this class, our goal is to think like mathematicians. One thing mathematicians do is make sense of problems and persevere in solving them. The way they solve them is by following a particular avenue of thinking. Our goal is to get good at all three avenues, because sometimes we might find a particular type of problem is more easily solved using one type of reasoning than another. We might start using one avenue that we’re typically more comfortable with, and then get stuck as we start to solve the problem. We might have to try a different avenue of thinking to get unstuck. Other times, we might successfully solve a problem using one avenue of thinking, but it doesn’t give us sufficient evidence to convince a skeptic, so we might need to solve a problem using multiple avenues of thinking to convince the skeptics.

As we navigate these three avenues for thinking, there are some important things to keep in mind. We need to be able to construct arguments using our reasoning and critique the reasoning of others, we need to be able to model with mathematics, to use tools strategically, and to attend to precision.

So these eight ideas set our goals for the year. A student who has mastered all 8 has achieved our goal for thinking like mathematicians. They’ve grown in a way that’s more comparable to the previous year as well – for example, maybe in 6th grade, they were struggling with looking for structure and making use of it, so in 7th grade, they’re going to focus on that avenue of thinking. And maybe by the end of 7th grade, they’ve started to master looking for and noticing structure, but they’re still not quite sure how to actually make use of it, so in 8th grade, they’re going to focus on using the structures they notice to solve the problems.

Now, in Grace & Amy’s book, they have an appendix with the following chart:

Screenshot 2018-07-18 02.19.52

You’ll notice that there’s clear questions to “ask yourself” for each avenue and there’s clear “actions” to take when solving problems using each avenue. Let’s focus in on what the rubric might look like now. I’m going to choose just MP7 about Structure to focus on, because I’m most familiar with that practice.

Here’s the CC’s text:

CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

The Ask Yourself Questions: The Actions:
·       What type of problem is this?

·       Does this remind me of another problem situation?

·       How is this (situation, object, process, etc.) behaving? Can I connect it to something else I know?

·       What are the parts (chunks) of the process?

·       How can I get the answer without doing all the calculations?

·       How can I use properties to uncover structure?

·       How can I change the form of this (number, expression, shape) to surface the underlying structure?

·       Chunk complicated mathematical objects (expressions, shapes, etc.).

·       Connect representations.

·       Change the form of the number, expression, space, e.g., create equivalent expressions.

·       Recall and use properties, rules of operations, and geometric relationships

The Ask Yourself Questions:

  • What type of problem is this?
  • Does this remind me of another problem situation?
  • How is this (situation, object, process, etc.) behaving? Can I connect it to something else I know?
  • What are the parts (chunks) of the process?
  • How can I get the answer without doing all the calculations?
  • How can I use properties to uncover structure?
  • How can I change the form of this (number, expression, shape) to surface the underlying structure?

The Actions:

  • Chunk complicated mathematical objects (expressions, shapes, etc.).
  • Connect representations.
  • Change the form of the number, expression, space, e.g., create equivalent expressions.
  • Recall and use properties, rules of operations, and geometric relationships

Now, I’m currently envisioning a few different ways students could assess themselves about how well they’re using this avenue for thinking.

One, you could put it into the same four-point mastery scale from my last post, though I do find the “above grade level” to be a bit challenging here. One thing we haven’t specified is that deploying this practice looks different at different grade levels (in part because of the types of content the students are exploring), but also because their level of independence with this may vary.

1: Below Mastery 2: Approaching Mastery 3: Proficient(At grade level) 4: Mastery (above grade level)
Rarely uses the ask yourself questions OR cannot make sense of the math problem through asking the questions.

Cannot see chunks or ways to change the form. Does not make connections between representations.

Does not recall or use relevant properties or relationships.

Does not result in successful solution NOR do they switch to a more successful avenue.

Sometimes uses the ask yourself questions to make sense of math problems.

Is sometimes able to successfully chunk, connect, or change the form to solve problems.

Recalls and uses only some of the relevant properties or relationships.

May or may not result in successful solution.

Regularly uses most of the 7 “ask yourself” questions to make sense of math problems.

Chunks, connects, or changes the form to solve problems.

Recalls and uses relevant properties and relationships.

Results in successful solution.

Regularly uses all 7 “ask yourself” questions to make sense of math problems.

Chunks, connects, AND changes the form to solve problems.

Recalls and uses relevant properties and relationships.

Results in successful solution that student can verify using a second avenue for thinking.

Gradeless Gradeless Gradeless Gradeless

Another way I could see assessing it is by using Rhonda Bondie’s “Must Have” and “Amazing” criteria. For example (and I’m not sure I’m in love with this way yet – it’s my first draft still):

Must Have Amazing
·      Ask yourself at least 2 of the “ask yourself” questions to make sense of the problem.

·      Chunks, changes, or connects.

·      Valid mathematical thinking shown that begins the problem.

·      Ask yourself at least 5 of the “ask yourself” questions to make sense of the problem.

·      Does at least two: chunk, change, or connect

·      Can justify why it makes sense to chunk, change, or connect in the way that they did

·      Recalls and uses properties, rules of operations and geometric relationships

·      Can use another avenue for thinking to solve the problem and verify answer.

If you needed to use a 4-point mastery scale, you could do something like:

1: Most of the Must-haves

2: all of the Must-Haves

3: All of the Must-Haves and some of the Amazing

4: All of the Must-Haves and ALL of the Amazing (or MOST, depending)

I’m not sure which of these two ways I like better yet, but I recognize they’re not quite equivalent. Either way, I can now see much more specific goal setting around mathematical thinking if students become comfortable with the math practices as avenues for thinking.

They might say “My goal is to use structure to solve at least two problems in the next unit.” or they might say “My goal is to use two avenues for thinking to solve every problem.” or they might say “My goal is to get better at asking myself these questions when I use this avenue.” or they might focus even more narrowly on something like, “I’m going to improve my ability to chunk problems, and I’m going to use that strategy on at least two problems.”

Now it even makes sense for a teacher to share from one year to the next a student’s goals. Well, Kit was very strong at using MP7: make use of structure, but he really struggled with reasoning abstractly and quantitatively. Encourage him to develop goals around MP2 next year.

I want to do more fleshing these ideas out over the course of the summer. This connect came organically out of a conversation I was having with Rhonda this evening.

What do other people think about these ideas? I’d love to hear from you on here or on twitter/FB!

Goal Setting in Math vs. ELA (Part 1)

A few years ago, I was in a meeting to discuss vertical alignment among the math departments at my middle school. We were looking to trace how the skills and standards built from 6th grade to 8th grade in our math classes. One of the members of the meeting was a special educator who worked with both the math and ELA departments, and she introduced us to a document that the ELA department had been working on where they looked at how the standards built vertically as well. In that moment, I was suddenly even more jealous of ELA teachers than I already was (sometimes, I think I should’ve become an ELA teacher – but that’s a different post!).

If you look at the ELA common core standards across the domains and grade levels, they build on each other very directly. I randomly chose Reading: Literature to look at across the three grades. I chose the first standard within that strand, and I followed it from 4th grade through 10th grade.

Refer to details and examples in a text when explaining what the text says explicitly and when drawing inferences from the text.

Quote accurately from a text when explaining what the text says explicitly and when drawing inferences from the text.

Cite textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Cite several pieces of textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Cite the textual evidence that most strongly supports an analysis of what the text says explicitly as well as inferences drawn from the text.

Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

Look at how connected these 6 standards are. In all of them, they reference using textual evidence, but they do so in more and less sophisticated ways across the grade levels. At first, kids are expected to just refer to details, then be able to quote specifics, and eventually they’re citing evidence – first any evidence, then several pieces of evidence, then choosing the strongest evidence, and ultimately, being strong and thorough in their citations. All of the standards mention both what the text says explicitly as well as drawing inferences from the text.

This makes vertical alignment pretty easy and it makes it more obvious when a student is above or below grade level in this standard – for example, if a student is only citing one piece of evidence and they’re in 7th grade, you can see they might be operating at a 6th grade level, whereas if a 7th grade student is citing the strongest piece of evidence, then they might be operating above grade level. I confess there seem to be some degree of subjectiveness on the part of the teacher about assessing whether the evidence cited is the “strongest possible…” But there’s no question in my mind that these 6 standards are linked directly, and that there is a growth in the student.

If I’m a middle school student, I can even do a self-evaluation, rating myself on a four-point mastery scale relatively easily. Let’s say I’m a 7th grade student. I might consider my skills along the following rubric:

1: Below Mastery 2: Approaching Mastery 3: Proficient

(At grade level)

4: Mastery (above grade level)
Quote accurately from a text when explaining what the text says explicitly and when drawing inferences from the text.


Cite textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.


Cite several pieces of textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.


Cite the textual evidence that most strongly supports an analysis of what the text says explicitly as well as inferences drawn from the text.


Grade 5 Grade 6 Grade 7 Grade 8

Now, admittedly, it’s possible that even as a 7th grade student, if I’m below grade level, I might not even be quoting accurately from the text. But, there’s definitely a progression of the skill, where I can self-assess “Where am I now?” and then “What’s my goal?” And this goal even hints at the above-grade level work for the following year, allowing a student to push themselves. If I’m struggling with the current work, I can even see perhaps where I should have been developing it – if I’m not able to cite several pieces, am I at least citing one? Am I at least quoting something accurately? If not, then I can set a SMART goal: “In the next book club chat, I will cite at least two pieces of textual evidence to support my inference.”

Let’s contrast that with mathematics for a moment. First off, it’s practically impossible to trace a domain by code the same way I did with ELA – from elementary to middle school, the domains change names, and in 8th grade, one domain is replaced with another! I chose to look at Numbers &  Operations in Base ten in Elementary and the Number System in Middle school, as they seemed to flow together. Again, I chose just the first standard in each grade level at this domain. You can see for yourself how much less clear the thread is connecting these.

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

What the heck?? It’s practically impossible to see how knowing there are numbers that are not rational is even connected to adding and subtracting integers or how either of those ideas are connected to fraction quotients. Other than the use of the words rational and integer in the RN.A.1 (which comes from the high-school “Number System” standards), I’m not sure that you can see how this standard builds on the middle school ones either.

They do a much better job at connecting the two from elementary school, but maybe that was because of the different domain name? 5.NBT.1 builds clearly on 4.NBT.1, because now in addition to recognizing that a digit in one place of a multi-digit number represents ten times as much as the place to its right, students will also understand that it represents 1/10 of what it represents in the place to its left.

There are other planning documents that show the progression of standards, as you can find here:

And Randall Charles does a great job at summarizing big ideas in mathematics which you can link to the standards:

Cathy Fosnot has place big ideas in the landscape for learning which also includes models and strategies.

All of this tells me there seems to be an extra layer to understanding the way math content is connected to the big picture that doesn’t seem to exist in the ELA standards. It has always made me wonder how helpful information from previous year teachers was in math, especially in middle school – for example, does a student’s understanding of how to add and subtract integers impact their ability to identify a number as rational or irrational? I think not. That’s not to say that the skills for identifying a number as rational or irrational aren’t laid down in earlier grades or carried through to later grades. Quite the contrary – you just have to know which skills are related and how this is all connected. No easy task for a teacher who’s busy mastering their own grade’s standards – hence the progressions, and this other ideas I shared here.

This brings me back to the meeting I was sitting in: vertical alignment within the math department. It’s harder to identify when a student is above or below grade level if you don’t know what your grade-level standard looks like in earlier/later grade levels. It’s harder to differentiate for a student if you don’t know or understand how the math builds over the years. And the arrangement of the standards in math does nothing to shed light on it.

I also couldn’t help but notice the common core’s website is differently arranged for math vs ELA – in the math section, you must click on a grade first (K – 8) and then on the domain, and then on individual standards, whereas in ELA, you needed to choose a domain FIRST, then a grade! That speaks to the idea of vertical alignment more than the discrete topics and standards arrangement that pervade the math standards structure – which also seems to indicate the way that many people seem to consider ELA skills to build on each other whereas many people often view mathematics as discrete topics or skills that are to be memorized (but are seemingly disconnected from each other).

Recognizing this challenge of the math standards made the idea of goal setting challenging. How can you set a goal and see where you were, where you are, and where you’re going without giving away some of the story in mathematics? If I want to develop a conceptual understanding of the Pythagorean Theorem using the areas of tilted squares on grid paper (thanks Betina Zolkower and CMP3 and MAP), then I need to know if the student understands how to find the area of rectangles, triangles, and tilted squares. I need to know if they have already heard of the formula a^2 + b^2 = c^2. I need to know what they know about triangles, specifically right triangles. Understanding the triangle inequality theorem may be helpful, though it’s not necessary. Once they know what it is, they may learn the converse of it, how to use it to find missing side lengths, and how to expand it to work in three dimensions. This connects to their work with exponents (what does that symbol mean? How do you “undo” it? etc.) as well as with solving equations (if you push them to work algebraically vs. working arithmetically).  Eventually, they should understand that the Pythagorean Theorem is generalized for non-right triangles as the law of cosines. They will learn about trig for right triangles eventually as well, and they might study special cases of right triangles. The Pythagorean Theorem is also connected to similar triangles and what they may know about angles of a triangle (including that the sum of the interior angles is 180 degrees). This theorem might also help them as they find the areas and perimeters of composite shapes involving triangles and/or inscribed and circumscribed triangles. They will also eventually learn how to prove the theorem – and they may be introduced to any of the 2000+ proofs that exist.

I bring all of these ideas up to illustrate that it’s much more difficult for a student to self assess where they are in terms of their prior knowledge, their current understanding, and what their goals should be in mathematics class. If I’m a student who is in middle school, just learning about the Pythagorean Theorem, how do I assess which aspect I’m struggling with, or how to go above grade level, or how to set a goal? I don’t have nearly as clear of an idea just based on looking at the standard or even if I looked at a rubric. I’m not sure I could even think about what a rubric would look like for this topic in the same way.

The standards about this theorem exist in 8th grade:

Explain a proof of the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
It’s talked about explicitly again here:
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
It’s the underpinnings for these two standards, but you need to already know the law of cosines is connected to PT to know that:
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
So let’s see: if I was an 8th grader, learning about the Pythagorean theorem, would I necessarily be able to use a rubric to understand where I was in my understanding? I’m not sure. Here’s one attempt at making a rubric for it…
1: Below Mastery 2: Approaching Mastery 3: Proficient (At grade level) 4: Mastery (above grade level)
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in mathematical problems in two and dimensions. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two dimensions. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

(NOTE: Big jump to understand what trigonometric ratios are!)

Grade 8 modified Grade 8 modified Grade 8 HS Geometry
When you look at this rubric, I tried to modify the 8th grade standard a bit to show partial understanding. I figure if grade level requires using it in both 2D and 3D contexts, then only being able to use it in the 2D context would be one example of approaching grade level. For level 1, I struggled with what to write, ultimately eliminating the real world problems, thinking this is basically a student who is just plugging and chugging numbers, but doesn’t necessarily understand what they’re doing or how to apply it to a context. (Though, I fully admit that a student who is below grade level in this standard actually might have even bigger challenges – they might not know how to square numbers or how to follow the order of operations or how to solve an equation with exponents – any of those skills could be preventing them from being able to apply this formula). For the above grade level, I tried to pull in the related HS geometry standard – but it mentions trig ratios, and those aren’t something that a student is going to be able to just “push themselves to do” based on what they know so far – in 8th grade, they may never have even heard of trig yet! (depends on their teacher, their school, etc.).  An alternate possibility in the above grade level section could be applying the formula on composite figures, inscribed/circumscribed triangles, or finding patterns in special right triangles.
This means it’s pretty hard to set goals and self-assess. How do I recognize if my difficulty is that I’m forgetting to square the numbers first, and I’m adding a + b and then squaring it? How do I notice that needs to be my fix? If I can use it in two dimensions but not three, I feel like that might be easier to recognize and set a goal around, but I (as a student) might not be sure HOW to figure out how to use it in three dimensions without outside support, whereas I feel like a student who already knows how to cite one piece of evidence can more easily push themselves to cite at least two (or even to choose the BEST evidence) relatively individually.
Today, in my conversation with Rhonda Bondie about goal setting in math, I had an epiphany. All this time, I’ve been trying to wrap my mind around having students set goals related to the math content standards. But they don’t flow well from grade level to grade level, and it’s not always super obvious how they’re connected. It’s hard to set a goal when you don’t have a clear idea about where you are, what your struggles or, or what your goals are. Feedback (from peers and teachers) also plays a role in goal-setting, as the feedback may cue you in on your strengths and struggles. But how do you give each student specific feedback every day in every content standard during the unit? So often, it feels like student goals (when I’ve had them set them before) are either focused on the “wrong” numbers – their grades (I’m going to earn a 90 on all of my quizzes, or on my exams, or on my report card), their HW completion (I’m going to do my HW every day), or their participation in class (I’m going to raise my hand more, or participate in class discussions more). I rarely hear/see my students set goals like “I’m going to learn how to solve problems with the Pythagorean Theorem in 3 dimensions,” – though occasionally, especially at the start of the year, I’ll hear a student set a goal related to math content from the previous year, like my student who said “I want to master long division because I don’t really feel comfortable dividing.” Sometimes, these content goals from the previous year are good goals (in this case, it was a sixth grader, so it was relevant!), but other times, those goals aren’t directly related to the new skills of the grade level (i.e. if an 8th grader said “I’m going to master how to do integer operations” – it’s an important skill for 8th grade standards, but it’s not in-and-of-itself related to any 8th grade standards!).
And then, in the course of my conversation with Rhonda, I had an epiphany about goal setting in math class. I’ll share that in my next post, so stay tuned!

Friendly Critic Observations

This year is only my second year in my current school & my second year choosing “informal” observations only. These are “drive-by” observations where my principal and AP stop by for about 15 minutes before passing on to another class. They send me a write-up with some feedback – but they rarely discuss these lessons in person – or in detail. On the one hand, getting all highly effective ratings makes a person feel like their hard work pays off – on the other hand, it doesn’t actually help me grow!

On Monday this week, I had someone come in to observe me and coach me a bit. She’s a former teacher who’s been out of the classroom only a few years, and is currently working on PD for teachers through the DOE. She followed me around when I conferenced with students, and we had a really good lens for her to view my class with; our lens for the day was balancing student independence with interdependence and how problem content, context and teacher questions can support this balance. I think that needs to continue to be my lens again next year because I felt like I was just scratching the surface in listening to her feedback.

Because she was not evaluating me – in fact, had absolutely NO POWER over me at all, I was able to listen and be present with her feedback. I felt like I knew she wasn’t judging me, but was supporting me to become a better teacher, so I could explain without getting defensive. Next year, she offered to come visit my classroom earlier in the year, and try to get in more frequently, so she can really provide me with some coaching – and I felt really good about that idea.

To this end, we asked, “What are some ways to increase student accountability for listening to each other during the whole group share (and not only to you to reiterate or confirm, implicitly or explicitly)?” And “how do you transfer some of the responsibility for moderating the group share conversation to them?”

We brainstormed a few ideas of what to do differently in the future, such as:

Turn-and-talks for vocab refreshers

Co-creating what they share out as per team vs. per individual (and thinking about how to encourage diverse voices when sharing on behalf of the teams)

Accountability for actually turning and talking (had one boy in a group of 3 who was totally silent all class)

Asking “What did you team notice/wonder?” vs what did YOU notice/wonder and building accountability around truly having a team response.

Clarifying questions and answers can be team based (at least as a 1st pass) to free me up to ask conceptual questions when I conference with students (which also means I need to have better conceptual questions planned)

Instead of answering the clarifying questions myself, ask the students,”Review your worksheet. What questions do you have about what to do?” and then direct them to spend the first 2-3 minutes of the explore portion of the class clarifying those questions with their table mates and asking, “Have you asked your partner?” Then, when I circulate, just listen in, instead of answering.

Follow Yvonne Grant’s advice of think of “What question can I ask” when students ask a question (instead of providing an answer).

Cultivating my “Kara face” (i.e. my poker face) when students say something incorrect. (I’m usually better about that, but in this case, it was a tangental topic, and I felt rushed, so I didn’t spend as much time clarifying as I could have).

I wonder if I can also view the student feedback from this same lens of the tension between independence and interdependence. I think I have a longer post about that alone percolating in my brain, but it’s not ready to come out yet.

Soliciting Student Feedback