My Speech is now live on youtube! For those of you who missed me talking at MfA’s MT-squared, here it is: Become the Subject.
My Speech is now live on youtube! For those of you who missed me talking at MfA’s MT-squared, here it is: Become the Subject.
Last week, I tweeted about one of the two restorative circles I ran in my two math classes that weren’t functioning well. I haven’t tweeted about the second class because there’s been more on my mind about it, and I haven’t been able to put it all into words. However, in honor of MLK day and several of his quotes, I want to share something that happened in my last period class during our circle on Thursday (and then my follow-up after school & on Friday/this weekend).
I started with the four agreements from courageous conversations about race:
I explained what each of those meant in kid-friendly terms and expressly said we weren’t “naming names,” but speaking about how WE felt during class. Then I asked the same opening question, “How does math class feel to you right now?” and I listened to the students. Because this was last period and their issues are different than my other class, the conversation went very differently. I had three students talk about DREADING school & many kids who spoke about feeling bored, burnt out or ready to go home from school at the end of the day. I asked them about what they found engaging in their lives and what would help keep their attention on math at the end of the day as our two follow up questions (different from the morning’s class who was having trouble with calling out and interrupting people making mistakes, who I asked how it feels when they make a mistake in math class & who they ask for advice). I heard the students of last period request more games, physical activity, and hands-on activities, as well as stories that were more “fun and deep,” so my Friday lesson attempted to incorporate several of those aspects to be more engaging for them. While it was obviously just a start, many students who were there expressed that it did feel different and fun and nice (some of the students who said they dread school had not come to school on Friday, supposedly fearing a math test – but that’s for another post).
There was one student in that last period class who spoke up whose contributions were very different. She mentioned that students had been laughing at her because I call her by a different name than the other teachers do, but that in her culture, it’s traditional for people to get two names, not just one, and so the second name I was calling her was actually part of her name and not just a nickname I had made up for her (which was what other students insinuated). But most teachers just read off the roster what your name is, and they don’t ask you what you want to be called, so the other teachers just called her by the first part of her name. She said she was sick of being teased and harassed about the name that I called her because other kids didn’t understand. Names and naming have been on my mind a lot lately, in part because of my own experiences with names, realizing I was mispronouncing a white student’s name earlier this year, and another student for most of last year. Plus, Dulce-Marie Flecha had just posted about pronouncing her name correctly on twitter (Duel-seh-Marie, for those who don’t know), so I was thinking about the importance of student names already.
I spoke with my student after class a bit more about it, and she expressed that she had students who were in 7th and 8th grade that she didn’t even know teasing her in the hallway, and I felt angry. I began writing an email to my colleagues to call her by both names, but I decided to stop by the 6th grade AP’s office, since I had just seen he was in. I told him a bit about what had happened, and he expressed that he thought it would be a better message coming from guidance, and that I should refer the situation to our guidance counselor. I told him I didn’t understand why, since I was the person this student had felt comfortable sharing this with, and that it felt like passing the buck to pass off the issue to the guidance department – and also, I didn’t understand why teachers would be more receptive to hearing this information from the guidance counselor and not from me, one of their colleagues.
The other piece that felt uncomfortable throughout the conversation was the fact that he suggested we talk to her parents and find out what name they wanted us to call her. I expressed the fact that it shouldn’t be necessary to ask parents – especially in the case of a trans student, that would be violating their right to privacy and self-determination (NYC protects a student’s right to be called ANY name THEY use and identify with, and doesn’t require parental notification of such, especially if it would potentially harm the kid). He pushed back & said that for all of the other students whose names differed from the names on their records, the request to use an alternate name came from the parents. I said that it didn’t have to and shouldn’t have to, that we should be asking the students about names and pronouns and not making assumptions. He expressed that she had never told any other teacher about this name difference, and I challenged “Which other teachers asked?” & he said something about her being capable of self-advocating. I said maybe, but why should she have to. I also expressed that not every parent felt comfortable or spoke English fluently enough to make these kinds of demands of the school, & he said he’d spoken to her parents about other (disciplinary-related) things and they were fluent in English and seemed capable of asking for that if they had wanted her to be called a different name. Our conversation didn’t seem to be moving forward, and he was distracted (and continually shutting down my attempts at engaging with “I still stand by what I said…”). I left with this unresolved, but continuing to chew on what he had told me.
He thought about it for a bit, and came to find me in my classroom a bit later to continue the conversation. I appreciated the fact that he acknowledged the conversation was unresolved, that he wasn’t feeling well and hadn’t been giving me his full attention, but now he could give the conversation more attention (though he was still feeling ill). We continued talking about the two issues. He mentioned the idea that people might be more receptive to hearing critical feedback from someone who didn’t seem like a “peer” but seemed like an outsider (i.e. guidance) because it would feel more accusatory coming from me. I expressed that I felt angry, like we had robbed this student of her name.
He also mentioned a fact that infuriated me (though I don’t remember now whether it was during the first part of our conversation or the second part). My school population is about 50% Hispanic students and has been for all of its 6-year history. This was NOT the first example he could think of a student who used two names at home that our school hadn’t “caught” initially, and that eventually (perhaps by 8th grade, where our curriculum focuses on self-advocacy) most students who wanted to go by a name different than the one on the register ultimately asked the school to do so (that he KNEW of! was the piece I thought but didn’t articulate at the time).
I pushed back on this assumption, and tried to make an analogy to my experiences in college using Kit, a name different from my legal name at the time (and even in K – 12 school, I never used my legal birth name, always a nickname). I expressed how every year, at the start of class, I had to tell my teachers to use a different name than on the register for me because I didn’t use that name. He expressed something like, “You should be proud of yourself for advocating for yourself like that!” and I snapped back, “No, I didn’t want to have to ask for that. It’s exhausting to constantly have to fight that battle. I wanted the teachers to ask me what name and what pronouns to use, and not just me – but to ask EVERYONE, so I didn’t stand out as different.” He tried to make a connection with the 8th grade curriculum teaching our students to advocate for themselves, and I said, “But that’s ridiculous. Not everyone is going to stand up to a teacher and tell them to do something different. We’re the adults; we’re the ones in power. And we have the power to change the system so that the names we use for students are the names they identify with. We shouldn’t be changing their names or making them ask us.”
He seemed to chew on this idea a bit, and said next year, things could be different, and we could discuss how at some of our Wednesday 6th grade meetings. He reiterated how he thought that the guidance department would be better suited to introduce this topic to the 6th grade team and recommended I refer the student to the guidance counselor.
I vented about the situation to my black colleague the next day, and she expressed no surprise, but sympathy for my position (and agreed with me at the absurdity of making students request something that teachers should be asking for as a norm). I also wound up seeing the guidance counselor during fourth period (after she’d had a chance to speak with the student) & I spoke with her about the situation. She told me the student had said she would prefer if I only called her by the first name since everyone else was doing that. She said that the kid had expressed that she had wanted to try out using both names (since half her family called her both), but no longer felt comfortable doing so at school and would just keep that other name for outside school/family. The guidance counselor mentioned that she had a similar experience (she is also Latina/Hispanic). The guidance counselor was having her lunch in the staff lounge at the time of our conversation (I ran into her on my way to the bathroom, which is in the staff lounge), and she was seated with the other guidance counselor and one of the office secretaries. I mentioned how I felt angry that we hadn’t created a system to ask students about their names, and the secretary shared the story of her son’s Greek name being Americanized in school (because it is spelled differently than it is pronounced, so teachers/students made assumptions about how to pronounce his name). They all seemed relatively complacent with this situation though – it felt like they expected it and they didn’t think there was another alternative (i.e. students having their names, even “foreign-sounding” ones or double-names used and respected by people at school, people OUTSIDE of their home culture). I mentioned that I wanted to try to change the policies and how we asked students about what name to call them, and I asked if the guidance counselor could come talk to the sixth grade department about the situation. I expressed to her what my AP had said to me (that the idea might be better received, coming from an outside source like the guidance department), and she agreed, though she expressed the fact that our meetings are always during her lunch (I said she didn’t have to come for the whole meeting if she couldn’t stay). She’s (supposedly) going to come talk to us during the Wednesday meeting next week; I’ll see how that goes.
I’ve been thinking about this all weekend (my partner has heard me yell about stealing a kid’s name from her no less than three times since Friday afternoon). I think I want to propose that next year’s “summer assignment” is for each student to video themselves saying their names and their pronouns, and then the first week of school, every student’s HW is to watch the other students videos and learn the names and pronouns of the students in their classes. That’s a WAY better assignment than the “flashy” one they do now with reconstructing a puzzle cube over the summer. (I plan to propose this on Wednesday, during our team meeting).
For me, sharing this experience about what happened last week honors MLK’s work because I now understand his Letter from the Birmingham Jail SO MUCH better because of this situation (especially these specific quotes).
The first is this quote from Letter from the Birmingham Jail, “I must make two honest confessions to you, my Christian and Jewish brothers. First, I must confess that over the past few years I have been gravely disappointed with the white moderate. I have almost reached the regrettable conclusion that the Negro’s great stumbling block in his stride toward freedom is not the White Citizen’s Counciler or the Ku Klux Klanner, but the white moderate, who is more devoted to “order” than to justice; who prefers a negative peace which is the absence of tension to a positive peace which is the presence of justice; who constantly says: “I agree with you in the goal you seek, but I cannot agree with your methods of direct action”; who paternalistically believes he can set the timetable for another man’s freedom; who lives by a mythical concept of time and who constantly advises the Negro to wait for a “more convenient season.” Shallow understanding from people of good will is more frustrating than absolute misunderstanding from people of ill will. Lukewarm acceptance is much more bewildering than outright rejection.” (Emphases mine)
The second is this quote, “I had hoped that the white moderate would see this need. Perhaps I was too optimistic; perhaps I expected too much. I suppose I should have realized that few members of the oppressor race can understand the deep groans and passionate yearnings of the oppressed race, and still fewer have the vision to see that injustice must be rooted out by strong, persistent and determined action.” (Emphasis mine)
I used to be offended by his statement that the white moderate was a stumbling block for the black stride for freedom, because I saw myself as a white moderate (though now I begin to recognize I might be more of a white radical…?). I thought that surely the oppression and hate felt from active racism like the KKK was far more harmful than the passive racism from the white moderate. I didn’t get it. Until now. I see it now, in this experience with my colleague (who has been involved in courageous conversations work in the school system, so he is not unfamiliar with “the work” to be done) and being furious at him for not recognizing how expecting the students to “advocate” for themselves when he/we (the staff/faculty/teachers/admin) have the power to CHANGE the system perpetuates systemic oppression of students whose “legal” first names don’t match the names they go by (for WHATEVER reason – whether it’s due to cultural differences of having multiple first names, going by a middle name, or being transgender and using a name other than the one on the birth certificate – or even just as simple as having a nickname that feels more true to ones’ self than their legal name).
I recognized that I see the structural solution to the “issue” of mis-naming students and misusing students’ names which he was blind to. The structure that we need to put in place is to disconnect the birth certificate (and a student’s “official” record) from the name we (the teachers, the students, and the school community) know a student by, and instead, use the name that the students themselves use for themselves. In order to do this, we can restructure how we learn students names at the start of the year to ask students to tell us what names to call them (and not just SOME students whose names we read and think look “different” or “unfamiliar” but instead to ask ALL students and make NO assumptions about whose names will be familiar for US to pronounce vs. unfamiliar!). This works for students of multiple cultures as well as for transgender students. It honors students’ identities and grants them the dignity of having their names (and pronouns) respected by their peers & by the people who hold the power in their school – the teachers. We can root out the injustice of robbing students of their names not just for individual students (and thus have to keep repeating this work for each individual student), but systematically, by changing the way we interact with our students and their identities. By shifting the paradigm of student names from assumptions that the name on the register is correct or the student will advocate for themselves to asking ALL students to name themselves for us and each other, we shift the labor from a small group of students who are other-ed by the restrictive system to the entire system and to ALL students, so that ALL are on equal footing when we enter school. No assumptions made about whose names we need to repeat or practice, no assumptions about whose names will be hard for US to pronounce, but rather, an assumption that everyone has a right to hear their name pronounced correctly and be called by the pronouns & name they identify with (and that the burden should rest on the community to unpack our assumptions and biases rather than on the individuals to “advocate” for themselves in an unjust system that privileges some “familiar” names above the rest – which reminds me of this video from Key & Peele about pronouncing names according to the way a name is pronounced within a particular culture – and feels like the Ann-ah vs Ahn-na discovery from earlier this year).
My colleague, had (has?) a narrow vision of supporting our students. Instead of acknowledging that the system is BROKEN and that the SYSTEM is oppressive, he would instead teach our students to “speak up” or “advocate” for themselves – not paying attention to the weight of that (constant) emotional labor. I realized in this moment that he didn’t (as a white, cisgender, heterosexual, man) understand how heavy it is on you to continually have to advocate for your basic human dignity. I realized this is one of the ways that the “moderate white” can act as a stumbling block on the path to freedom – instead of using the power he has to change the structure of how our school learns the names to use for students, he instead has perpetuated this systemic oppression without even recognizing that there was an alternative. He told me that for 6 years this school has known that students from Hispanic backgrounds often have two names, and yet there is STILL no structure in place to ask ANY students what name we should call them.
I am beginning to learn the hard truth behind Dr. Martin Luther King Jr’s quote, “We know through painful experience that freedom is never voluntarily given by the oppressor; it must be demanded by the oppressed.” I hoped that the people at the school I currently work with (who SEEM to be open to conversations about race & inequity) would better see how to support our students, how to free our students, but instead, their vision of freedom is only within the current oppressive system, rather than re-envisioning an entirely new system/replacing the parts of the system that are broken. It seems like their vision is still limited; we’ve got more work to do. I get the sense that they’re open to doing this work (some of them, at least), even if they don’t know how. As I was bringing up the issues of trans students to this AP, during this conversation about names (and making the connections to my own experiences with names and pronouns in college and our students with multiple names), he mentioned that he had never really done any research about trans issues and wasn’t as familiar with them. I mentioned that two years ago, when I came out as trans on-stage at my school’s annual GSA assembly, my principal commented to me afterwards that she learned she was “cisgender” that day. He said I should be proud of that and making those changes, but I said that it wasn’t my job to educate her about that. She should’ve been doing that work on her own – and I recognized in this conversation what my fellow colleagues who are Educators of Color experience on a regular basis. People who are well-meaning and congratulate you for educating others, not recognizing their privilege and power in the situation means THEY should be one the ones educating themselves and advocating for people with less power than they have (both centering those people’s voices when it’s relevant, and helping to bear the emotional load of advocacy when possible). Shortly after this conversation with my AP, I saw the threads on twitter about something that seemed too similar (I can’t recall now what I posted this in response to, and it’s no longer attached to the thread it came from… Maybe it was about the incident this weekend with the Native Elder & the MAGA bigots?), but I tweeted, “It isn’t the oppressed’s responsibility to educate oppressors. It’s the responsibility of allies with privilege & power to use it to educate others who are (unknowingly) supporting oppression. Placing that responsibility on the shoulders of the oppressed perpetuates oppression.” And I recognized the analogy to what happened in my school there in tweeting that. So I’ve been considering what my role as a teacher with privilege and power is towards my students and my colleagues.
I’ve begun to realize how I’ve been complicit in white supremacy by not better educating myself about the history of racism and oppression in this country, by not having read more Dr. Martin Luther King, Jr, more James Baldwin, more Audre Lorde, more Maya Angelou, more authors of color. I’ve said before (including in a facebook post recently) that I cannot remember a SINGLE black author whose NOVEL we read IN-class (and then analyzed, etc.) from grades K – 12!! And I remember A LOT of the books we read, including the Joy Luck Club (by an Asian-American author, at least one non-white author?). I don’t even know that I ever read a book by an African until college, when I took a human resilience class (taught by my first black teacher! Senior year of college!!) and we read several memoirs by black people (including the books, Warriors Don’t Cry and A Long Way Gone). I do remember reading excerpts of books (like Other People’s Children by Lisa Delpit) in other college classes by Black authors (and other Authors of Color), but I had such a white-washed upbringing, despite growing up in Queens (one of the most diverse counties in the country!). Now, thanks to the #DisruptTexts movement, I’m way more intentional about seeking out a variety of stories in my media consumption, so I don’t fall into the trap of the “single story.”
My reflection on this experience last week and how it relates to the teachings of MLK bring me to thinking about a different quote that I first heard as an undergrad in college from the Office of Service Learning. The office was run by two people who seemed to truly understand that “community service” couldn’t begin and end with a one-time contribution, but that service to the community required understanding the community, becoming a part of the community, and figuring out what the community needed from you in order to support them. The quote was this, “If you have come here to help me, you are wasting your time. But if you have come because your liberation is bound up with mine, then let us work together,” from Aboriginal activists group, Queensland, 1970s. I see this quote as connected to a different quote from MLK’s Letter from the Birmingham Jail, “Injustice anywhere is a threat to justice everywhere. We are caught in an inescapable network of mutuality, tied in a single garment of destiny. Whatever affects one directly, affects all indirectly.” (Which, as an aside, I’ve included in my family’s Social Justice Haggadah, inspired by my first Social Justice Seder with Rabbi Mike in college).
For me, it was recognizing the oppression and injustice that I face as a transgender and queer person is tied up and connected to the oppression and injustice that people of color face (in my school’s case, the Hispanic students, in MLK’s speeches, the black people of America). While it is uncomfortable to admit, I was very much raised with the “white savior” complex (thanks to parents & ten years in Catholic school in Queens, NYC), and that influenced the ways I saw some of my students in the past. It’s unfortunate that it took me feeling the weight of oppression myself to understand others’ oppression, because if that’s what’s required to feel empathy, then we’re in trouble. I have begun to realize that it is my job as a white educator to support and push my colleagues and my students in doing this work of unpacking our biases and recognizing systems of oppression (and how we can be complicit even while not thinking of ourselves as racist). It is my job as a white educator to amplify my colleagues’ of color voices & ideas, and to decenter myself and my own experiences.
I have begun to grow through my intentional expansion of my teacher twitter network. Although I originally “insisted” it was only going to be for math educators and following math education, once I began following the José Luis Vilson & Marian Dingle, I quickly realized that there’s no way to keep social justice out of my education and still honor my students’ identities and dignity. From my conversations with them, as well as through following EDUColor on twitter, the #DisruptTexts twitter chat run by Tricia Ebarvia, Lorena German, Dr. Kim Parker, and Julia Torres, and #CleartheAir twitter chats, run by Val Brown, the resources from Tolerance.Org & voices like Shana White, Julie Jee, Absurdist Words, Ebony Elizabeth Thomas, Clint Smith III, Chris Emdin, Tsɔɔlɔ Awo, Kelly Wickham Hurst, Chanda Prescod-Weinstein, Nikole Hannah-Jones, Melinda D. Anderson, Dr. Kim Tallbear, Dr. Debbie Reese, Leslie Mac, Cornelius Minor, Sara K Ahmed, and so many more than I can name in the time I have tonight (if you’re not directly following BIPOC on your social media/news consumption, though, you need to diversify the voices you listen to and amplify), I’ve begun to recognize the work I still need to do to be anti-racist in words AND in actions. I’ve begun to distinguish between my intentions (which may be “good”) from my impact (which may not be good for ALL people), and analyze both to make sure I’m having the intended impact. I’ve watched allies like Christie Nold & Jess speak up on twitter and through their blogs (in particular, thinking about thinking about the amazing work I’ve seen Jess do with her 5th graders on her blog, Crawling out of the Classroom, and shifting the lens to inquiry about whose stories are heard and whose voices are absent). I’ve begun to be more aware of what (systemic) racism looks like, and I’ve begun to use my voice to disrupt the status quo, challenging people (students, educators, family, tweeters, etc.) who are ignorant to the impact of their actions or inaction. The more work that I do, the more I recognize how much more work there is to be done (on myself included). I’ve learned to question my own assumptions, my own biases, and my own blindspots and complicitness in racism.
I’ve read and thought about twitter threads like the following about whiteness & equity work in schools & how to stop being complicit and start being a disruptor:
I encourage you to seek out voices like these and listen to them, think about them, and begin to do this identity work on yourself, ESPECIALLY if you’re an educator, so you can then support your colleagues and students in unpacking & examining their assumptions and biases and their impact in the classroom. Here’s a link to the next website I’m going to be reading about how to dismantle white supremacy in our schools. I’m doing some learning right now about understanding Culturally Responsive Teaching & the Brain by Zaretta Hammond (reading the book & taking an MfA course on Building & Rebuilding Authentic Relationships that references it).
I shall end this post with one of my favorite quotes from Rabbi Tarfon, also quoted in my Haggadah, that I find reassuring when I am feeling overwhelmed by oppression and how far we still have to go. This quote originally appeared in the Pirkei Avot (Chapters of Fundamental Principles) as part of the Mishnah (Jewish Oral Law), and simply states, “It is not incumbent upon you to complete the work, but neither are you at liberty to desist from it” (Avot 2:16).
I’ve been a part of several different professional development sessions this year that have spurred my thinking about the distinctions between equal / equality and equivalent / equivalence with regards to math.
It started brewing in my mind because my school’s math department wanted to investigate how we develop the notions of ratios and proportions across the grades and how we develop the notions of equations and expressions across the grades – and we discovered the ideas of equivalence and equality came up in both discussions, so we expanded our view-point to see RP & EE as applications of equal vs equivalent.
I also have been thinking about the lesson from Illustrative Mathematics’ 6th grade curriculum entitled equal vs. equivalent: https://im.openupresources.org/6/teachers/6/8.html
In that lesson, they use these words to distinguish between equations which are true for ONE value (because the two expressions are EQUAL at that particular point) from equivalent expressions where they are ALWAYS true at any point. With regards to equations, it becomes clearer to see that if both sides of an equation are ALWAYS true, we would say it has infinitely many solutions. We could relate it to the graph of two equivalent lines and see that they would result in the same line.
In thinking about solving an equation algebraically, though, I can also think of creating EQUIVALENT equations (by using the properties of equality) where we simplify the equation on both sides to eventually isolate the variable, and discover the singular value that makes the equation true. For example, the equations 24 = 3(2x – 4) and 6x – 12 = 24, and 8 = 2x – 4 and x – 2 = 4 and 2x = 12 are all equivalent equations (meaning the same value of x will make them true). With some prior knowledge of algebraic manipulations, one can see how multiplying, dividing adding and subtracting can create those equivalent equations, which have an equal solution. However, one might not consider the equation 5x – 4 = 26 to be equivalent, since it’s less obvious how to transform the above equations into this one, however, if we consider “all equations with the same solution to be equivalent” then it would be, since x ALSO equals 6 when that equation is true.
However, the idea that my expressions are equivalent (instead of merely equal at one point) comes up when we apply other properties, such as the distributive property, the commutative property or the associative property, to show that various expressions are equivalent. For example, the expression 3(2x – 4) is equivalent to the expression 6x – 12 because they are the same for ALL values of x. If we “set them equal” and wrote 3(2x – 4) = 6x – 12, the equation would be true for any value of x.
From these examples and discussions, I began to see a distinction between when two expressions are EQUAL vs when two expressions are EQUIVALENT.
Then I started thinking about ratios and proportions and equivalence. We would say that the ratio 2:3 is equivalent to 4:6 or 40:60, but we might not say those ratios are equal, because they are NOT identical and they are not the exact same, though they describe an aspect of the relationship that must remain the same. Even with fractions (whether they describe a part-whole or a part-part relationship), I can think of two different fractions as being equivalent, but I’m not sure we would describe them as equal, even though we might use the equal sign (2/3 = 4/6 = 40/60, but those are equivalent fractions, not EQUAL fractions).
We also might say that there are equivalent ways to represent fractions in decimals and percents, but I’m not sure if we would describe THOSE as equal either! For example, if I have the fraction 2/5, I could also describe it as 0.4 or 40%, but are those equal or equivalent?
This made me wonder if there are ANY times when we can think about ratios and proportions from the EQUAL sense, rather than just noticing equivalence among ratios.
Kara Imm led a PD last Wednesday with “Meeps” and “Bleeps” that made me think about this distinction more deeply. On an imaginary planet, 32 meeps had the same length as 24 bleeps. We discussed that this meant that the ratio of Meeps to Bleeps is 4 : 3, and that we could even say it with fractions: 1 : 0.75 or 1 1/3:1. But when it came time to write the equations to convert from meeps to bleeps, many people fell into the “reversal error” and switched where the numbers went, saying that m = 0.75b instead of b = 0.75m, which made me recognize that there were two distinct quantities being counted/measured here: the NUMBER or AMOUNT of meeps and bleeps and the LENGTH (or size) of meeps and bleeps. In our ratio table, we had been considering the ratios of the NUMBERS of them, but when we drew a ruler to show the lengths and an equation to convert from one to the other, we needed the variable to represent the AMOUNT of them, so that the expression 0.75m would be equal to the length of a bleep or 1 1/3m would be equal to the length of a meep.
It made me have a new understanding and appreciation for why it’s so tricky for people to distinguish HOW we’re using the letters (as variables or as units), and how we’re using the equal sign – as a symbol of equality or as a symbol of equivalence!
I also made a connection between these ideas and geometry, how in geometry, the “objects” themselves (the angles, line segments, etc.) are only ever said to be either congruent or similar to another one, but the MEASURES of those objects (the measure of the angle or the lengths of the line segments, etc.) are equal. I wonder if that’s a helpful distinction here as well…
I also started thinking about the classic “coin” problems where you talk about the number of coins you have and the VALUE of those coins and try to find a solution… Writing a system of equations for that is often a challenging problem, because the coefficients of each variable in the “value” equation is implied by the type of coin used.
So now I’m wondering whether it EVER makes sense to discuss ratios as being equal, or if they’re only ever equivalent. What are other people’s thoughts?
A couple of the links that I read that pushed my thinking as well:
I saw this on Facebook today, and it really set me off: (I actually really dislike the accompanying article’s explanation, as I think it still perpetuates the problem instead of ACTUALLY developing the depth of understanding necessary):
Problems like this are why I appreciate the Boss Triangle & the NCTM’s article, “The Problem with PEMDAS.” It drives me nuts when people try to talk about math as being arbitrary based on examples like this. What I REALLY think these “viral math problems” illustrate is a lack of numeracy and a lack of understanding of the relationship between the mathematical operations AND an over reliance on mnemonic devices/acronyms to remember procedures… but, those mnemonic devices actually mask the inherent relationships necessary to figure it out (because, as Pam likes to say, math is figure-out-able).
If you understand division to be “multiplying by the reciprocal,” and you convert the divison by 2 to be multiplying by 1/2 (and thus .5), then there is literally no question that the value is 9.
You get 6 * .5 (1 + 2) as your new expression, and you now have flexibility in how you evaluate it: whether you start by taking half of 6 (which is 3) & then multiplying it by the sum in the parentheses (also 3), the product of which is 9, or you distribute the half to the two terms in the parentheses and get 6 (.5 + 1), which is 6 (1.5): still 9. There are no longer possible paths to get 0, 1, 3, or 6, which are the common “wrong answers” when trying to evaluate the original seemingly ambiguous expression. (And, if a mathematician WANTED you to divide by the PRODUCT of 2(2+1), they would have used a vinculum, often known as the “fraction bar” and put that whole expression in the denominator, with only the 6 in the numerator – grouping symbols also help mathematicians to be less ambiguous I. their intentions).
The REAL problem here is that people do not develop a deep enough understanding of the relationship between the operations to understand the ORDER of the operations. And instead of taking the time to fully develop those relationships, we fast forward students through relationships and to procedures.
The order of operations is NOT arbitrary, it is based on two key relationships:
Addition & subtraction are inverse ops & Multiplication & division are inverse operations (and radicals & exponents are inverse operations), and thus you can rewrite any radical, division, or subtraction as its inverse operation. For subtraction, use the additive inverse with addition or for division, use the multiplicative inverse (often called the reciprocal) with multiplication. You can even write radicals with fractional exponents: in this case, you indicate the inverse operation by using the multiplicative inverse of the exponent (i.e. root 2 is with an exponent of 1/2, which is the multiplicative inverse of 2, which is the power you’re trying to “undo” by taking the square root).
Thus, in considering the order of operations, you actually need only consider exponents, multiplication & addition. This is step 1 for deepening your understanding, because it explains why “PEMDAS” should actually have the MD and AS on distinct levels of the hierarchy: they are equivalent to their inverses in the order because you can eliminate division and subtraction by replacing them with M & A.
Secondly, the operations are prioritized in order of power: more powerful operations are done first (hence exponents are done before multiplication and multiplication before addition). I define powerful here to mean the operations that would change a number more if the specific numbers in question were kept constant: for example, ab will change a more than a+b, and a^b will change a even further (certainly for the integers, and even the rationals, but I think this is true for all real numbers). (I use “change” in a large part here to mean the difference between your sum or product and the original a).
I like explaining this relationship as being about power, because then it holds true even with the rationals, but students with less experience with the operations may find it helpful to also consider the fact that exponential notation (with whole numbers only) can be rewritten as repeated multiplication, and multiplication (of whole numbers) can be rewritten as repeated addition (of whole numbers). I hesitate to ONLY use that (even with kids who have less well developed relationships between the operations), because this type of understanding of the operations limits a more in-depth understanding of rational number arithmetic (i.e. addition, subtraction, multiplication, division, and exponentiation of numbers with fractional/decimal/negative parts). But that’s a whole separate issue.
Grouping symbols, like parentheses, often get thrown in to the order of operations as “coming first,” but actually, that is not at all a correct or complete understanding of the meaning or use of grouping symbols!
I also think it is confusing that we use parentheses (and brackets and braces) to group expressions together (within) and imply multiplication between quantities (one inside and one outside the parentheses), as people have to first ask themselves whether the parentheses mean multiplication or “group this expression together and treat it as a single quantity” – or worse, both meanings simultaneously!
Now, the truth is, grouping symbols are NOT required to be dealt with “first,” rather, it is that the terms/expression contained within a grouping symbol must be treated as a single quantity rather than each part of that expression treated as separate quantities that are “free” to be operated on by other parts of the overall expression.
That’s why you need to either simplify the expression inside the parentheses and then multiply OR use the distributive property to distribute the “divided by 2” (as a factor of 1/2) before you can do anything to the 1 and 2 inside the parentheses (in the original expression that inspired this post today). This could bring me to a rabbit hole about how we don’t deal with quantities that are not single numbers well, but that’s a whole different post! For now, I will simply say that experience with growing patterns and treating “chunks” of the visual as a being represented by a single can support the development of this type of understanding.
Which brings me to the last point about the order of operations: you can use the properties of numbers to make equivalent expressions: associative, commutative, distributive properties… all of which apply to MULTIPLICATION & ADDITION (but not division & subtraction), which is the other reason why if you convert the division and subtraction to multiplication and addition, the expression becomes easier to evaluate (and not as ambiguous). Both subtraction & division specify a left to right order inherent in how we record them (we do something differently to the subtrahend and the minuend or the divisor and the dividend), but because addition and multiplication are both commutative and associative, we simply call ALL of the chunks of them by the same name: factors in multiplication and addenda in addition.
These “viral math problems” are only ambiguous to evaluate if you DON’T understand the RELATIONSHIPS between operations and instead rely on acronyms and mnemonic devices to remember “what to do.”
I think if we spent MORE time on developing an understanding of these relationships between numbers and relationships between operations, we wouldn’t need to “teach” the order of operations at all: it would be a logical conclusion to these questions. It would be “figure-out-able.” In fact, if I were to sum up what I’ve learned from my work with Pam Harris, I think it directly connects:
If we develop a deep understanding of RELATIONSHIPS & QUANTITIES, then math is figure-out-able.
(And, typically, the development of that understanding comes from repetition of a relationship or a quantity & analyzing the patterns seen in the reptition OR looking for structure within quantities or relationships>> now I’ve connected the three avenues of thinking from Amy & Grace to Pam Harris’s work).
Oh, I also just felt some sparks connecting my ideas from this to the ideas that numbers represent either quantity or a relationship & the ideas of “math as a second language” & the ideas of adjective-noun that were recently shared with me on twitter… but that connection will have to wait for another post.
So I am doing something new and unusual at my middle school this year. My school has school-wide intervention periods three times a week (we call it cerebral diversity or CD for short). CD is right after lunch, and the class is in smaller groups than a normal class for MOST of the groups (I have 12 students in mine). Students are sorted (somewhat) based on their state exam scores from the previous year – students who are at or above grade-level get to take “electives” like physics, school blog, or debate, while students who are struggling readers often are placed in intervention groups like “Just Words,” “Wilson Reading,” or “Small Group Reading Instruction.”
There are only two of us doing math intervention. The other person is trying something new this year – he’s pulling students out of other groups (primarily reading intervention) for 6-weeks at a time to use Math IXL for students to practice skills they need.
I did a completely useless intervention last year with some online program, and I don’t think any of the kids moved in their landscape of understanding or in their test scores. So I decided to try something new and different. This year, I confess, it’s a bit rocky and there’s no over-arching unit plans yet, because I’m making it up as I go along. I always take the “three-year view” with new curriculum though: the first year is a complete hit or miss mess because you don’t know what you’re doing yet, the second year is when you start getting the gist of the unit plans and order of the skills correctly, and the third year is when you really refine and hone the individual lesson plans. Even then, I think it’s really a good 5 years before you become proficient at teaching a course (this is why I think every teacher should teach the exact same course 5 years in a row, but I digress).
So I’m in my first year of this curriculum, and I’ve never had the opportunity to design a class like this before – so I’m super excited! I think it’s actually going to be profound, both for me (in the long run, having developed this class, I can then share it with other teachers in my school and/or other teachers/schools in my city or even around the country, I suppose – though it’s focused specifically at sixth graders who enter middle school without fluency and proficiency in math).
I started out with a few basic goals that I outlined in my “unit plan:”
Year-Long Essential Question
|How can I set goals and grow as a mathematician?|
|How can I make sense of math problems like a mathematician?
How can I notice, describe and analyze patterns?
How can I reason abstractly and quantitatively?
How can I look for and make use of structure?
How can I look for and express regularity in repeated reasoning?
How can I participate in a community of mathematicians?
|Math Cerebral Diversity is where targeted mathematics instruction lives. Each CD should aim to:
I had some ideas right from the start of things I wanted to include and incorporate. The first thing was that I know I want to reinforce some of the routines I use with the whole class in math instruction, where I have the students notice and wonder, or I have the students notice, describe, and generalize patterns. I knew I wanted to use games, though at the beginning, I didn’t know which math games I would use (I’ve now built up some of a list). I also knew I want to emphasize vocabulary (unit-specific and math in general), as well as support the skills/topics we’re doing in “regular” math class AND support their numeracy (my bigger focus).
Some of the routines I either plan to use (or have already begun to use) with these students: notice/wonder, three reads, problem strings, I have/you need, the routines for reasoning, rumors, and one I call “fancy date.” I also plan to use lots of math games, and I have started to grow my list: “How Close to 100?” from Jo Boaler (a dice-rolling, area game), I have/you need?, and today, KenKen. (I also plan to use something called “find the Factor” /twisted tables eventually, but I haven’t gotten there yet). I realized that Kenken’s whole advantage is that it supports students in practicing number facts (but in a puzzle/fun way!), which is BASICALLY the goal/point of this course for me. So today, I taught them how to play Kenken in the most basic of basic ones – just addition and just a 3 by 3 and a 4 by 4. Tomorrow, I’m going to give the kids MORE practice with the KenKen puzzles, and I’m going to start administering the MRI interviews tomorrow while the kids work on the new puzzles (since they can work semi-independently).
What’s the MRI, I hear you ask? I discovered (through twitter) Marilyn Burn’s Mathematical Reasoning Inventory. I’ve had students complete the written portion (I was absent for a day, and I needed sub plans!), and now I need to perform the interviews with the kids to assess their reasoning and strategies for solving the arithmetic mentally. Since there’s 12 kids, it’ll take me some time to interview them all, but I figured now that they know a puzzle/game that they can be relatively independent on, I can start conferencing with them individually/in small groups to gather the data.
Ultimately, I would love to be able to assess where the struggling students are along Cathy Fosnot’s Landscape of Learning (for addition/subtraction, multiplication/division, fractions/decimals/percents, geometry, and algebra, and statistics >> but I don’t know that those all EXIST yet!), and then track how they grow over the course of the year. I’ve accepted that this part is probably a year 2 or 3 goal; I’m not sure yet how to gather this data, and I haven’t been doing it thus far!
Now, the other component that I haven’t talked enough about is Pam Harris’ work. Maybe you’ve heard me post about her lately? I have been taking one of her online courses (this one is about developing powerful multiplication), and it’s totally influencing the way I think I’m going to design this class moving forward this year AND how I’m going to restructure it for next year.
First, I introduced the game “I have/you need” last week and did a little bit of formative assessment – one of my students even asked for it today! (we ran out of time, so we’re going to try it again). I noticed from that game that two of my students didn’t know their “pairs to ten” yet – and many of them struggled with the pairs to 100 that weren’t “easy” (heck, I even struggle with that – that’s why I’m so much slower at mental math than my partner – she knows all of those pairs to 1000!). I had heard about this game before, but Pam’s website nicely laid out exactly how to begin and how to structure this game so it gets progressively harder and so that the kids start to build fluency with some of the facts.
A second thing I’ve been learning about (from her multiplication online course) is the different strategies for multiplication. She has 5/6 strategies that she says are VITAL for kids to know to be able to efficiently solve any problem that’s reasonable to solve without a calculator. The strategies (as I’ve understood them so far from her) are basically broken into two types: the ones that use the associative property (and maybe also the commutative property) and the ones that use the distributive property. She has named them doubling and halving (which can be extended to tripling/thirding, etc), and then flexible factors (where you can actually change the numbers you’re multiplying by rearranging the factors – for example, 6 x 35 can become 21 x 10 because 6 = 3 x 2 and 35 = 5 x 7, so the 5 and 2 can be regrouped and the 3 and 7 can be regrouped, and the 21 x 10 is MUCH easier to do mentally!) and then there are three types of chunking: “smart” partial products (to be distinguished from the four-part “place value” chunking prioritized by the algorithm), 5 is half of ten, and over/under (where you go to a friendly number that’s a little above or below your target and then adjust with addition or subtraction). Once students have the idea of a fraction as an operator, you can also use “quarters” as a strategy (this would mostly be an associative property one), where to multiply by 25, you could divide by 4 (or halve and halve again) and multiply by 100. She mentioned that some of the strategies come sooner on the Landscape for learning, and that would also support kids in mastering them first (it gives us, as teachers, an idea of the “usual progression” through the ideas, so we can support students in moving forward into more efficient/sophisticated strategies as they get stronger with the concepts and skills).
So then tonight, Pam also mentioned two big ideas (plus a bunch of littler, supporting ideas) in the class that I want to incorporate into my planning for CD:
First, the associative property-based strategies are easier for the kids to own because it “stays” in the multiplication whereas the chunking requires the students to know WHAT they’re adding or subtracting (which, when we keep it in a context like packets of gum, they can make sense of), but doubling is lower on the landscape of learning, a chunk of which is shown above, so they’re more likely to master that first. So this made me start thinking about the best order to introduce the strategies (via the various strings). I tried to do a x9 strategy with my kids the other day, and they seemed like they were starting to get it in the moment, but when I tried to revisit it the next day, they hadn’t retained it. And when I shared that with Pam (and Kim and Sue, and a few others at the end of tonight’s course), they reminded me that of course students hadn’t integrated it yet! It was brand new to them, and in the same way that I wouldn’t expect students to have mastered anything else after only doing it once, I need to do problem strings to develop a particular strategy more than once with my kids as well! Which brings me to a related idea, which is that we want to order the strings so that they’re developing the strategies in a manner that the kids can master them – but also so that the kids get enough experience with the “type of strategy” (and both the ratio table and array models) to be able to “own” the strategy and the ideas behind it. All of this reassured me that they will eventually master the x9, as long as I keep revisiting it. But it also made me realize I should use the landscape of learning to my advantage & start with an easier strategy to master, like double, double, double.
The other thing we discussed tonight were the “basic facts” and how to have strategies for each of them. She even sort of went in an order that seemed like it could be the order that I introduce it to the kids. Here’s what I heard.
To support our students learning their multiplication facts, we need to focus in on one type of fact at a time, developing strategies for that number. She highlighted in color as she went through. My understanding of the order (for the 1 – 10 facts FIRST) is that kids should understand what it means to have “zero groups” of something >> and then what the zero facts are. They should next understand what it means to have “one group” of something >> and then they need to own the one facts. (I think that might be part of my conversation with the kids tomorrow, before we start playing Kenken, after we’ve played “I have, you need”). Then, they can move into doubling (x2), doubling and one more group (x3), and double-double (x4). From there, she suggested skipping to double-double-double for (x8). Then we went up to the x10, which is a place value shift. I was actually surprised in some ways that we had waited so long to get there, but I realized that noticing the place value shift is harder than just skip counting by tens or “adding a zero,” so I actually think it makes sense to develop some multiplication understanding before going to the times ten – because now, I could see an array that was n by 1 being repeated ten times to show the 10n. From there, she moved into the idea that 9 is 10 groups minus one group and then that x5 is half of times ten.
These strategies (combined with recognizing the symmetry in the table and that ultimately, you can “reverse the order of the factors” without impacting the product – the commutative property) reduce the number of facts you might need to “memorize” down to three facts: 6×7, 6×6, and 7×7. For some reason, she said 7 x 7, the kids often seem to know, but then we could think about how else to support kids in learning these facts. And I discovered my new favorite for 6 x7. If you know 3×7 = 21, then doubling the 3 to get 6 means we can double the 21 to get 42 (and 21 is pretty easy to double!). We can actually continue that pattern to do 12 x 7, where we can see that we’re doubling the 6 to get 12, so we can again double the product, which again is pretty easy – 42 doubled is 84. So now we’ve not only gotten the ten-times table, we’ve already started thinking about the 12 times table.
I’m guessing that the “elevens” are going to be using chunking of 10 groups plus one group (sort of like the “opposite” of the nines).
The 12s, I can see in a few ways:
I actually just thought of another “game”/routine we need to do in my CD. We should do the choral count-arounds (or other ideas) or whatever they’re called. I just thought about how powerful it would be to get kids practice “doubling.” What if we started with a number, and I said “15. Now double it.” and then pointed to kids in our small circle, one after the other, and they had to double the previous number (in this case, 15, 30, 60, 120, 240, 480, 960, 1920, ahh!) instead of just doubling for powers of 2!
This is basically my attempts at consolidating some of my learning from tonight’s multiplication session, and to connect it with the work I’m trying to do in my CD. I’m also simultaneously “planning” for tomorrow and this year!
Oh, I just remembered another thing Pam mentioned – she talked about how to use flash cards (and now I’ve gotta check and see if I can find my deck – I hope I didn’t get rid of them!). Use them to categorize whether a student “just knows” a fact or if they don’t have it mastered yet (split the deck into “mastered” and “not mastered” in an interview style). They’re not racing, but any that they don’t know in about 3 seconds, put in the “not yet” pile. Then go back through and make a “clue card” based on strategies they know. For example, if they struggled with 8 x 7, maybe they want the reminder “7 = 5 + 2” or maybe they want the strategy “double-double-double” or maybe they want to try “double 7 x 4” because they know that. They’re building on relationships they know.
There was also the practice with connecting the multiplication problems on the paper (instead of using a “mad minute” with timer and doing as many as you can!), and I really liked that idea too!
I think my goal for my students is to start by simultaneously developing their pairs to ten and pairs to 100 (to support their additive reasoning and move them from counting), as well as to develop their multiplicative thinking, by mastering these strategies AND by mastering their facts. Because the COOLEST part (imo) is that a kid who has a strategy like “5 is half of ten” now owns not JUST the first 12 multiples of 5, but ANY multiple, because they have a generalizable strategy that ALWAYS works. So if I say “13 x 5” they can do 130 / 2 = 65. If I say, “24 x 5” they can say “240/2 = 120.” Now yes, this requires them to have mastered doubling and halving – which is why I think this brings me back to where I’m going to go next.
Tomorrow, we’re going to do more KenKen puzzles & I’m going to start doing assessments of the students. I’m going to do as many interviews as I possibly can & I’m going to try to record the ones that students know/don’t know and/or their strategies (I haven’t decided whether I’m going to start with the MRI from Marilyn or just the multiplication facts like Pam suggested. I’m going to try to record where kids are for the addition pairs while we play a game of “I have, you need” to warm-up immediately after lunch). On Thursday, I’ll probably do the same exact thing so we continue until we finish the MRI/interviewing of the kids.
Once I have that mastered (probably by next Tuesday), I’m going to continue developing their pairs to 100, but also shift to focusing on the multiplication facts. I’m going to start with ensuring that we’re solid with a rule for x1 and x0 (I THINK we are, but I am making NO assumptions anymore!). Then, we’re going to work on doubling until we have it mastered. We’re going to work on doubling both with strings that will show us double, double = x4 and double double double = x8, and also doubling to build on x3. I’m wondering how to support kids in “mastering doubling” though for kids who don’t have that skill yet (Pam? Any thoughts?). That may be more of an “elementary” skill, and maybe some of them have it more than I think, but I also know the “pairs to 10, 20, 100, 50, etc.” will make the doubling easier. I also wonder if I might need to go into compensation for addition strategies, or if I should just leave that be for now…
Once we’ve mastered doubling, we’ll work on the chunking for times three – double and add one group, double and add one group, etc. I also think at this point, it’ll be important to practice halving – though I’m not sure exactly what to do yet here (especially, since I had a few students who struggled to recognize even numbers that were three digits or four digits!). I also want to show some of the strings of doubling and halving here, so the kids recognize that sometimes, it might be easier to make a problem into a related but easier problem (i.e. 18 x 5 is kinda yucky, but 9 x 10 is easy – and you’ve essentially taken a factor of 2 from the 18 and given it to the 5 >> which KINDA reminds me of the compensation strategy from addition, where if you were adding 18 + 5, which is kinda yucky, you might take 2 from the 5 to make the 18 into 20, and then do 20 + 3 instead).
I think once they know how to double and halve, we’ll move into 5 is half of ten (because they already have some fluency with the times ten and we discussed it in terms of dollar bills the other day), and then ultimately, we’ll revisit the 9s and 11s, thinking about the chunking strategies at that stage.
I want to shift the kids away from skip counting, and I want them to start thinking multiplicatively. I think this order might get us a pretty good start. Let’s see how we go.
Finally, the last thing I was thinking about was something that came up after the main session, in my conversation with the smaller group of folks who stuck around (yes, I’m THAT kid! No, you’re not surprised!). I mentioned how we had done a string for the 9s strategy on Wednesday last week, and when we came back on Thursday, they didn’t remember the strategy at all. And one of the teachers reminded me that it takes repeated exposure to the strategies before the kids own them, and I was like “Oh, duh! I know that.” But it helped to remind me that the kids need more experience with thinking about the relationships and making sense of the math, and not just being expected to catch up “all at once.”
One last thing (and this was influenced by Grace & Amy, so let me give credit, where credit is due). Their routines (because of the annotation on the posters) wind up leaving a tangible “remnant” that can be used as an anchor chart. They also have the routines end with a meta-reflection, so the kids think about what they saw today as being connected to a future problem. The prompts often say things like,
“When doing <a>, I learned to pay attention to…”
“When doing <a>, I learned to ask myself…”
“The next time I do <a>, I will…”
That might also help them retain more, because it tunes them in and helps them think about the critical parts that relate to future problems.
Alright, that’s it for real now.
I was rambling about my small group intervention to my partner, Charlie, tonight, and it was suggested that I should blog about this because it’s important to capture these ideas in writing. So here goes.
I’ve been thinking A LOT about designing a small group intervention in math that really support kids in the transition from elementary school to middle school. In particular, I’m focused on my “cerebral diversity” group – essentially, a group of 12 students who I meet with three times a week for about 35 – 40 minutes for additional math support. I’ve been trying to think about the best ways to support them in developing their mathematical reasoning.
So that brings me back to Pam’s diagram from my last post (this is clearly stuck in my mind!):
In middle school, I want students to develop their proportional reasoning (particularly in 6th grade), so that means they need a solid foundation of additive and multiplicative thinking strategies. I’ve already started to make connections to this diagram in my observations of my students’ struggles. Yesterday, I had a student do division by drawing circles and counting (one by one!) 64 tick marks into the 8 circles (and he only knew to draw 8 circles because he saw that number on his multiplication facts handout because he doesn’t have fluency with his multiplication facts). I realized he was stuck in a counting strategy for a multiplication problem – he didn’t even have additive strategies in that moment!
So this brings me to what I did with my small group intervention this week:
On Tuesday, we previewed the idea of common multiples by looking at the 2, 3, and 5 times tables. We talked about the numbers that appeared in MORE THAN ONE table and named them common multiples.
On Wednesday, we worked on a problem string to develop a strategy for doing “9 times anything.” We started by using money to talk about place value and multiplying by 10 and 100. I asked students to consider how much money I had if I had 14 $10 bills. And then I asked them how I could trade that in so that I had fewer bills. After a few people said “and two twenties” I clarified that the bank I went to ONLY had 1s, 10s, and 100s. We then filled in our place value with 1 hundred, 4 tens, and 0 ones. We did the same thing for 27 tens and then for 27 hundreds. Some of the kids had never heard the phrase “twenty-seven hundred” to describe “two thousand, seven hundred” so we discussed that language.
Once it seemed clear they understood how to multiply by ten (and had a better sense of WHY we shift the place value than the often heard “add a zero” explanation), I decided to move into this string. We started with 6 x 10, 6 x 9, and connected the two. Then we did 8 x 10 and 8 x 9. We did 7 x 10, then 7 x 9, and then made a connection between the two. I recognize now that I wasn’t using a good model to help the students think because I was just writing multiplication sentences up there. I also wasn’t grounding us in a context, so when I jumped to 14 x 10 and 14 x 9, they got a bit stuck. There were two answers in the room for 14 x 9: 131 and 126. We used the context of gum packets and sticks of gum (which they were all familiar with from class number strings I’d done with them) to reason that it must be 126. We continued to 13 x 10 and 13 x 9 and then 17 x 10 and 17 x 9. When this problem came up, one student shared how he did the subtraction as 170 – 10 – 7, and I asked if I could represent his thinking on a number line. He agreed, and my other students saw how his strategy was helpful.
I tried to scale up this strategy to anything times 99. I had the kids multiply by 100 and then one of them even predicted we would do x 99! Only one kid was actually able to use the strategy of x 10 – x1 successfully, and got 1683.
I also noticed that some of the kids were easily doing the x10 but getting stuck on the subtraction – and I realized they might not have subtraction strategies that were helpful. So I decided to focus on supporting that with the “I have, You need” game that Pam introduced me to.
On Thursday, I tried to revisit the strategy from the previous day for multiplying by 9, and most of the students had forgotten it! I asked them how to do 19 x 9 using the strategy from yesterday and only after much prompting a few kids were able to remember we did 19 x 10 and 190 – 19. I then asked them if they could write the “helper problems” for 62 x 9 (getting 62 from one of the students) and some of them could – but not all. I decided to try to have the kids generalize, but they struggled to say the two steps were “multiply the number by 10, then subtract that number.” At one point, I said “let’s say we have ANY number, let’s call it N” and I totally lost a few of them. I realized I was trying to shove something on them they weren’t ready for, so I pivoted. Instead of doing my original plan (which was going to be to build on the x9 strategy and try out x19), I decided to play the game with them instead.
We got in a circle, and I told them I was going to start small so we could learn the rules of how this game was played. I wrote “Target = 10” and I drew the following on the board: _____ + _____ = 10. I then told them I was going to give them a number (and pointed at the first blank) and they were going to give me the pair that we could add to it to make 10. I said the numbers in a somewhat random order, 9, 8, 6, 7, 5, 3, 1, 2, and I pointed at each of the different kids. I noticed two kids were struggling with this. One of them even said he wasn’t quite sure how to play, so I filled in the first blank with the number I gave him (7), and asked him for the second number and pointed at the blank. He had to count it on his fingers.
He shook his hand “so-so” when I asked if everyone understood how to play. I said ok, let’s try it with 100, and I gave the kids ONLY multiples of ten. Most of them were pretty good, but it became clear two of the kids were counting (I hope by tens, but it was silent, so I wasn’t sure). I moved into the multiples of 5 (so giving a kid 85 and expecting 15 back), and when I tried to give one student 25, she couldn’t get 75. I tried to relate it to money for her, but while she knew there were 4 quarters in a dollar, and if I had one quarter, she would have 3, she didn’t know how much money that was worth. She could say one quarter was 25, and two quarters was 50, but froze up when asked for three quarters.
This gave me great formative assessment about where the kids were. Since we were out of time, I told the kids that they should play this game with the types of facts they didn’t know yet (I mentioned to my two counters that they needed to work with the pairs to 10 first).
As an aside: in talking with my principal & AP, they’ve talked about how the kids don’t “move” in math – in other words, their test scores are stagnant. And I felt like this experience really showed me why. In 6th grade, the students are being assessed on their multiplicative reasoning and their proportional reasoning – but if they don’t even have additive strategies yet, they’re going to struggle with that level of reasoning! So I need to support my students in getting OUT of counting and getting to higher levels of thinking.
This leads me to my ideas about designing this support curriculum. I am thinking through the three days a week I have with these 12 kids, and I’m thinking about how to arrange it so that we have one day that’s supporting their work in class (either previewing vocabulary or a skill or reinforcing something, etc.), one day that’s a problem string with multiplication strategies, and one day that’s supporting their additive reasoning and moving them out of counting. I think we can play this game with target to 100 and once I get to know the students, I can differentiate the level of complexity I can give them based on which skills I see they need to work on.
As an aside: tonight, in the conversation with my partner, it became apparent why she’s so much faster with mental math than I am: she “owns” all of the addition pairs from 1 – 100 and 1 – 1000! Whereas I own 1 – 10, and 1 – 20, I’m shaky on my 1 – 100 pairs. I own the multiples of ten (i.e. 10 + 90 or 40 + 60), but when it comes to my multiples of 5, I don’t “own” the facts close to the middle. For example, I know the 95 + 5, 85 + 15, and 75 + 25 forwards and backwards without thinking. But 65 + 35 and 55 + 45 trip me up every time. I want to say 65 + 45 and 55 + 55, and I don’t want to use 35 at all there! When I get into the “ones,” it’s even worse! I don’t know them with fluency at all!
So I started practicing them tonight! And I’m going to keep practicing them, building the level of complexity until I have all of the numbers to 100 mastered! I’m wondering whether I should share with my kids that I’m working on these too. I think there’s something powerful for them to recognize that even adults might have gaps in their fluency, but that once you recognize a gap, you can practice that skill and improve.
Additionally, I now know that next year, I want to use this “game” earlier in the year to assess who’s still counting and who’s got some of these facts mastered.
At TMC NYC last week, Melvin asked me how I started with routines and which routines I would recommend doing first. It got me thinking about the evolution of using routines in my own instruction and so here is my long awaited reflection on his questions.
The first thing I will say is that even now, I’m still not perfect. I definitely still struggle to be consistent in my implementation and to reinforce these routines regularly. I find myself continually having to remind the students (and myself) of some of these norms and expectations. I hope eventually this year, as the kids learn these routines, it will become more clear and automatic – but with 6th graders, I’m not holding my breath!
It was when I started taking courses with Rhonda Bondie that I started considering the power of NAMING the moves of a routine so that we could be consistent, and “refer back to our play book.” If I say “that team just did a hail mary!” you probably have SOME vision of what that means, even if you don’t know much about football. In the same way, we can name OUR in-class routines with names so that if we tell students “do a Share-Check-Discuss,” they know what that means – in terms of their roles and responsibilities, in terms of the action pattern, etc.
And so I think it’s very appropriate that my routines list starts with incorporating simple ideas from Rhonda:
Confirm & Contribute – when students are sharing (either with partners or groups or even as a whole class), and we’re writing (a list, a table, etc.), I ask students who are “audience members” to listen to the idea and compare it with their own list in their NB. If they’ve got something similar, they should check it off (confirming it) and if it’s a new idea, they should write + before recording the new ideas in their notebook (contributing it to their list). This accomplishes two things: 1. it gets students listening and comparing their ideas with what’s floating around the room and 2. it gets students who didn’t come up with as many ideas on their own a list to refer back to because they’ve now copied some classmates’ ideas into their NB.
Listen for Patterns and Surprises – I pulled this out of the “talk-think-open exchange” which Rhonda does (and I never have), where I tell students what to listen for as they’re sharing something, and then I give them time to discuss with their groups a pattern or surprise they noticed. This helps them know what to listen for (and to try to make connections between the ideas they hear) and it always gives us something to share out.
While I like Rumors (in theory and with adults), I’ve rarely actually used it with students because I never seem to know HOW to deploy it. It’s a place where we should be able to share ideas around the class, and kids get to get up and talk to lots of different people, but I’m not sure that I always find a tremendous amount of value from it, as sometimes the kids just swap cards (without reading or talking) or they just try to go to as many people as possible without listening to the new ideas. So I’m not 100% I would include Rumors in my “regulars” list, but it’s in my “favorites with adults” to use.
I also like Rhonda’s Idea Carousel, but I haven’t actually used it in my classroom as a teacher yet, though I’ve experienced it as a learner and loved it! I also used it as a facilitator of PD, and I found it extremely helpful and useful as well.
I also created a routine similar to hers that I call “Share, Check, Discuss.” It’s purpose is to check work the students have already completed independently or at home and to save only the questions with major confusion for the whole class discussion. The way it works is this: in groups of 4, students choose one person to go first and a direction to go around. They each have the paper they’re checking and their pencil. The first person reads aloud their answer to number 1; the listeners either check it off that they agree OR they circle it that they disagree. They rotate, going through the whole page, just reading answers (no comments or questions at this stage). After they get to the last question, the next person should ask “What questions did people circle?” This will direct their conversation back to the problems people disagreed on. I will give each group a post-it note to record ALL of the numbers of the problems ANYONE disagreed about and then to STAR the problems they still want to go over. I use that data to choose to either go over the problems that the most groups requested OR to know if we can skip going over the handout’s answers.
So of those five routines, I’ve already taught my students the first two, and I haven’t used the last three with my students at all.
Then, I find there are some more math-specific routines that I’ve started to incorporate, like Notice/Wonder and Which One Doesn’t Belong? In my actual enactment of these routines, I utilize routines and moves from Rhonda – for example, in sharing out from a notice/wonder, I cold call students and I offer them the opportunity to confirm or contribute ideas to our list to hear more voices. For Which One Doesn’t Belong, I’ve had students move to different corners of the room and share in groups based on which section they first thought might not belong, and then I’ve had them prepare a reporter to share out the most common reasons or to share out any surprises. Alternatively, I’ve done domino discovers to ask students to share out their ideas about the WODB or even to share out the most important noticing or the wondering they’re most curious about as a group. Domino Discover is another routine from Rhonda.
I’ve done all three of those in my class this year already (WODB, N/T, and DD).
I’ve been moving from minor routines (some of which I would almost name “moves” over “full blown routines”) to more structured, whole-class routines that follow a bunch of stages, flowing from a launch into repeated patterns of individual think time – partner shares – and whole class discussions into a final reflection. I learned about this general structure from David Wees, based on the icon he has shared from the New Visions website.
For me, my first experience with these kinds of routines came from Kara Imm’s number/problem strings during the developing mathematics year long course at MfA. I explored Pamela Weber Harris’s books as well, and eventually enacted these routines, but I had trouble at first, because I didn’t have this framework in mind. Once I incorporated more partner talk, I found my flow in enacting these number talks/strings. Additionally, when I incorporated the meta reflection at the end, I found students actually took something away from those number strings that was concrete, so I found that amplified the power of it. The other support that’s not highlighted here, in this sequence, but I first really thought about from looking at David’s PPT slides was about the role of sentence starters in supporting student conversations about the math AND consistent use of iconography to indicate the roles, routines, or other constraints of a situation (such as pencils or no pencils, alone or in partners or in whole groups, looking or talking, etc.). He claims it’s out of laziness, but I actually think the consistency of that is helpful for students – now they know, if they see this symbol, this is what I should do in class!
The first two routines I learned that were this degree of structured were Contemplate then Calculate and Connecting Representations from Amy Lucenta and Grace Kelemanik. I’ve also learned the 3 Reads Strategy and I’m teaching myself the Capturing Quantities routine (because I think it builds nicely off 3 Reads, especially for 6th grade math). I learned CthenC first, but I actually like Connecting Reps MORE. In part, this is because I want to use CthenC to highlight interesting/specifically helpful strategies, but to select which students I will call on, I need to really hear their strategies – and that doesn’t always happen succinctly. That can mean that it takes me too long to select students. Whereas in Connecting Reps, I can ask them which one they connected, and know that I can ask them clarifying/probing questions while they’re presenting to highlight features of the connection they might not initially mention.
So far this year, I’ve done problem strings with my students to develop their multiplicative thinking. I haven’t yet done CR or CthenC this year, though I used both of them last year. I just launched 3 Reads yesterday with my kids for the first time, because it was the first time we really had a word problem to read through and make sense of. I hope as it becomes more routine, the students will get better at it. I had to pause with Capturing Quantities (and I’m not 100% sure I feel confident actually deploying it yet), but hopefully, I’ll try it out during my ratios and proportions unit.
I try to let my math goal determine which routine I choose, but I also know that sometimes, I’m thinking about my student “study skills” and “active participating” routines (like notice/wonder or confirm and contribute). I hope this list is helpful.
Let me first make a connection to Sara Van Der Werf’s post about being an evangelist: I have come to realize that I am a problem strings evangelist (and thus to some extent a Kara Imm & Pam Harris evangelist, as I learned basically everything I know about problem strings from the two of them).
On Wednesday last week, I had the amazing opportunity to finally meet Pam Harris in person! She came to my classroom and led a problem string with my class! (can you tell I’m EXCITED?!) Then we (and my co-teacher) debriefed the lesson, and talked math, math teaching, and so much more for the next few hours! I designed a string with her then, and I implemented (most of it) on Thursday in my classes. That’s a different blog post that I haven’t written yet; the focus of this post is about this image that she shared: both on twitter and in our conversation.
Essentially, this image represents the types of strategies we have for various problems. It does impose a sort of hierarchy to the thinking, recognizing that proportional reasoning, for example, is more sophisticated than counting strategies. I then watched this video where she explains it in even more detail.
(As an aside, as I watched her video, I thought about my own strengths and weaknesses as a mathematician and my own strategies. I realized that I have pretty decent proportional reasoning because my multiplicative reasoning is good. But my additive reasoning sucks. If you ask me to add a few numbers, I don’t have as much fluency as I should. And subtraction? Oh that can be a nightmare too! But then, I heard in that video I shared above that Pam mentioned the game “I have, who has?” and knowing the additive partners to 100… and I was like, “OH SHIT. I need to do that game!” Because recognizing the other piece out of 10, then out of 20, then out of 100 – that’s how you start building your additive reasoning strategies! But again, that’s another post for another time!)
This got me thinking about how to teach my 6th graders to find the LCM & the GCF. My school uses CMP3, a problem-based math curriculum. [Aside: I do think it is very strong in many ways, though I also think it has some gaps and weaknesses. My goal this year is to think through some of the problem strings that would support my students in developing the reasoning they need to engage in these problems. I can’t even tell you how excited I am that Pam’s going to support me in making those strings and the connections/integrations within the problems/investigations! Again, that’ll probably be a future post!.]
CMP3 uses three problems to introduce students to the LCM & GCF: the ferris wheel problem, the cicadas problem, and the snack packs problem. For those of you not familiar with the curriculum, let me summarize these three problems:
In the Ferris Wheel problem, they talk about different sized Ferris wheels that complete their revolutions at different times, and therefore, the two people who get in at the same time, go past the bottom at different times… So the students are asked to calculate when the people would next both be at the bottom at the same time: this turns into a common multiples problems. They’ve selected the numbers in the problem strategically, so that you’re finding the least common multiple of 20 and 60 (which is one of the numbers because 60 is a multiple of 20), the least common multiple of 30 and 50 (which share the factor of 10, so it is 3 x 5 x 10 = 150), and the least common multiple of 11 and 20, two relatively prime numbers (so the LCM is their product).
In the next problem, the students are taught about cicadas which have prime number life cycles, in that there’s a brood of these insects that emerge every 13 years and another one that emerges every 17 years. The students are then asked to find the next time both will emerge in the same year. Because the two numbers are relatively prime (and happen to be prime!), the students can just multiply 13 x 17 to find the answer. There’s an extended problem where they have to find the next time a 12, 14 and 16 year cycle would overlap as well, which is definitely a bit of an extension at this point, as it requires noticing that while 12 and 16 share a common factor of 4, the 14 only shares a common factor of 2. This problem also pushes students to try to generalize a method for finding the LCM for any pair of numbers and predict when the LCM will be equal to the product or less than the product of the two numbers.
The third problem shifts gears and asks the students to break up food into snack packs with the same number of each types of snack (apples and trail mix bags). The numbers involved are 24 and 36, which share a common factor of 1, 2, 3, 4, 6, and 12, so students are both recognizing common factors AND directed towards the greatest common factor. After this problem, students are asked to consider when real world problems ask them to use multiples and when real world problems as them to use factors. The hope is that students see that breaking two things into equal groups will require factors, whereas two things that happen repeatedly in cycles will involve multiples.
The third investigation in the book goes into “number (factor) strings” where they push kids to move from the product of two factors equaling a number to three factors, four factors, and so on, breaking the factors down through a puzzle to eventually find the prime factorization of a number. My new school’s unit plan basically skips over the prime factorization investigation because of the combination of time and the fact that it’s not “directly” related to the standard being assessed (i.e. students aren’t asked for the prime factorization of a number in 6th grade in NYC; they’re asked for the LCM or GCF of two numbers, either in context or not, and Prime Factorization is just one strategy they can use to find those values).
However, I’m now even more worried about how to run these three problems so that my students leave with (a) having made sense of when to find factors vs. multiples, but even more importantly (b) having made sense of HOW to find common multiples & factors with strategies BEYOND JUST listing multiples and looking for common ones. Now that I have Pam’s graphic in my mind, I recognize that skip-counting is a form of additive thinking (as they are technically repeatedly adding the number to itself as they produce each multiple), and my students need to move into proportional reasoning this year – so I can’t leave them in additive strategies. So I began brainstorming the multiplicative strategies for finding the GCF & LCM of two (or more) numbers.
I started by analyzing my own strategies for finding the LCM & GCF, and everything I could think of EITHER required knowing the prime factors of a number, involved skip counting, or (in one case) involved the Euclidean Algorithm to find the GCF and then the “LCM algorithm” (I don’t know if it has a more official name), which says for to find LCM (m, n), do mn/gcf(mn) (which is why relatively prime numbers have an LCM equal to their product – when you divide by their GCF, you divide by 1, and it doesn’t change the quantity). I wasn’t sure which (if any) of these strategies would be most accessible to my students and yet simultaneously enhance their thinking beyond just counting strategies.
So that led me to my twitter network to ask people how THEY think/teach about the GCF/LCM.
#MTBOS #msmathchat what strategies do you have for finding the GCF or LCM of two numbers? I am trying to anticipate what my 6th graders might try… and what I can do to support them in moving beyond listing multiples (my anticipation of the “main” strategy).
— Kit (@MrKitMath) October 4, 2018
And I got only a few different types of responses. I noticed there were certain trends: people used primes (either with factor trees or Venn Diagrams) or people used an “organization technique” like the ladder method/grid method/cake method. The only other thing that came up was Howie’s video.
First, I think I will try to find a way to incorporate Howie Hua’s video with my students, because I think this is a method for finding the GCF that involves subtraction and to some extent, repeated division (and actually builds towards the Euclidean Algorithm). But I do worry that it might leave students stuck in a lower level of thinking.
— Howie Hua (@howie_hua) September 16, 2018
One method people shared for getting the LCM & GCF of two numbers was using prime factorization and then organizing the prime factors somehow (often using Venn Diagrams) to find the GCF & LCM. People had different ways of organizing their prime factorization (often the “factor trees” method), but ultimately this method was as follows:
This is definitely one way to find the GCF/LCM efficiently – but considering that my school is skipping the Prime Factorization unit of Prime Time, I’m not sure whether or not my students will have the ability to use this reasoning/strategy – plus, many of my students don’t have mastery of any divisibility rules (for some of them, I even mean recognizing even numbers as divisible by 2!), so finding prime factors may not be the best method to start with. (Plus, if we were to follow CMP3, Prime Factorization comes AFTER common multiples/common factors, so theoretically, there should be OTHER methods).
The other main method that came up is called the grid method, the ladder method, or the cake method, based on the model that we use to organize the thinking involved. It all seems to be identical MATH, but different names for an almost identical “organizing structure.” Matt Coaty shared this video to explain this method. Essentially, you look at your two numbers for common factors, and you keep dividing your quotients over and over again until the two quotients from your two numbers are relatively prime – they you know you’ve got no more common factors to “pull out” and thus, you’ve found the GCF. From the GCF, you can multiply the GCF by the two “left over” factors to get the LCM.
Watching this video was a big painful, though. The narrator/teacher mentions how to multiply for the LCM, and it made me cringe so hard. She first doubles, doubles, doubles and doubles to get 16. Then says “you might not know how to do 16 x 7 in your head, so you can come over here and do work on the side. She then demonstrates doing 16 x 7… without actually showing any of the steps that she took to get 112! So I couldn’t help but wonder why she bothered to write anything down on the side at all! Then she does another thing I hate which is “string multiply” where she begins the next problem attached to the first problem (this teaches such horrible things about the equal sign and interferes with kids’ ability to understand it as a symbol meaning “equivalence” rather than a symbol meaning “find the answer” which is what most kids think when they first come to me in 6th grade. And finally, she gets the 560 without showing any steps again! Whereas (because I’ve worked with Pam), I started with 5 x 2 = 10, 7 x 10 = 70, and then I doubled, doubled, doubled to get 70, 140, 280, 560!
I realized that from watching this video, when they showed the L-shape, and she started multiplying from left to right, 2 x 2 x 2 x 2 x 7 which got her 112 (the original number on the left) and then x 5, and I realized she got the original number on the left from the first factors, and the only thing that was new was the 5. Instead of RE-doing the multiplication, she should have realized that 2 x 2 x 2 x 4 x 7 = 112, because that was the original number that it came from, and the only “additional” factor from 80 is the 5, so you need only multiply by 112 by 5. This then made me realize that if you figure out the greatest common factor, divide one number by that, you can take the quotient (the “extra factor”) and multiply the other original number by it. In this case, if you figure out that the common factor is 16 and then divide 80 by that (maybe halve, halve, halve, and halve? to get 5) then you just need to multiply the other original number (112) by that 5 to get the LCM of 560.
This made me think about something that I don’t think has been mentioned as much in the work of Cathy Fosnot’s “Landscape of Learning” & the work that Pam Harris is doing with secondary strategies. The landscapes always include strategies, models, and big ideas – and Pam has a great blog post distinguishing the difference between strategies and models. But I think models (“representation of a strategy”) doesn’t quite cover everything else, so I almost think there’s ALSO “organizing techniques” (Maybe this is a subset of models?).
Let’s consider the strategy from the video above, sometimes called the grid method, sometimes called the cake method or the ladder method. Essentially, the actual strategy says:
The model (organization technique?) most people use to represent this strategy is a grid that looks similar to a ladder, a cake, or an upside down division bracket. People have extended the Ladder Method to also apply it to factoring and using the distributive property, even with variables. (Now that I started calling it an organization technique, I’m wondering if it really is just a “model” – but not a model that can be used for other strategies? Or can it??).
Viewing this grid as a table made me think of some work from the CMP3 PD I led on Friday. We were looking at investigation 4.2, and considering the changing dimensions of two adjacent rectangles with a shared side.
If you look at each row, they can easily make an equivalent expression which represents the sums of the two areas (36su + 48su):
If you notice, the last row shows the greatest common factor on the outside, and the two numbers inside ( ) are relatively prime (3 and 4 share no common factor other than 1). This table could be seen as similar to the grid method, in that each time, the two numbers inside the parentheses were divided by the same factor. The main difference here though is that each row of the table “stands alone” because the factor outside the parentheses kept changing “in real time” and one subsequent row didn’t necessarily follow from the row right before it (i.e. you probably didn’t change the 2(18 + 24) into the 3(12 + 16) >> instead, you probably went back to the original and restarted with a different common factor). Regardless of which factor(s) you found first, you will still end in the same place with the greatest common factor being the 12 that gets pulled out.
I’m not 100% sure yet how this table integrates into the methods for finding the LCM/GCF of two numbers, but it FELT related (though, obviously, it builds on the finding of the GCF, because this is how you “factor” an expression!).
So this whole time, I’ve been feeling weird about the fact that in CMP3, you START with the LCM, and THEN find the GCF… but, in thinking about methods for finding the LCM, you always need to already know the GCF, or else you might just be finding a COMMON MULTIPLE, and not the one that is LEAST. When I went to Illustrative Mathematics’ open up curriculum, they reordered it, so that you find GCF first and THEN LCM! They then separate out the skill of actually USING LCM & GCF to a third lesson.
Of course, in those lessons, it doesn’t teach any particular organization strategy. Mostly, IM expects students to list factors and multiples (probably based on their prior knowledge) and then to circle the common factors or multiples and look for the greatest/least ones. Which (now) leaves me wondering if that’s actually supporting students in growing their thinking strategies, or if it just leaves them stuck in additive thinking strategies when they should be progressing to multiplicative thinking strategies.
I think if I were to classify the different techniques:
I think that students in middle school should be moving towards mostly multiplicative thinking, as that’s the necessary groundwork for proportional reasoning, which we need to start this year. So that would make me lean towards finding a good model to represent the division-related strategies, rather than pushing my students towards listing (which is the “previous level’s” thinking). However, I also wonder about imposing a particular organization technique on my students – because let’s be honest, that’s REALLY what the grid method is about. It’s a way to organize your division attempts to collect the factors you’ve found.
Which leads me to why this week’s lessons are still feeling “up in the air.” I don’t think I have made a decision about (a) the order that I’m going to teach GCF/LCM in, or (b) the techniques for finding the GCF/LCM that I’m going to teach. I also need to get all of my classes “caught up” on the number strings from last week – some of the classes got through 4 strings, and some of them only got up to one or two of the strings! I want to make sure all of the students get all of those strings, because I think they develop important strategies in the strings (that will ultimately help us in finding common factors and multiples as well!).