Day 1 and Week 1

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

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

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

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

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

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

Screenshot 2018-07-23 13.28.38.png

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

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

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

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

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

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

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

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

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

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

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

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

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

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Goal Setting in Math (part 2)

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

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

avenuesfrombook

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

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

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

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

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

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

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

Screenshot 2018-07-18 02.19.52

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

Here’s the CC’s text:

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

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

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

·       Does this remind me of another problem situation?

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

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

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

·       How can I use properties to uncover structure?

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

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

·       Connect representations.

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

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

The Ask Yourself Questions:

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

The Actions:

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

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

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

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

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

Does not recall or use relevant properties or relationships.

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

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

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

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

May or may not result in successful solution.

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

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

Recalls and uses relevant properties and relationships.

Results in successful solution.

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

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

Recalls and uses relevant properties and relationships.

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

Gradeless Gradeless Gradeless Gradeless

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

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

·      Chunks, changes, or connects.

·      Valid mathematical thinking shown that begins the problem.

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

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

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

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

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

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

1: Most of the Must-haves

2: all of the Must-Haves

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

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

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

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

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

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

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

Goal Setting in Math vs. ELA (Part 1)

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

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

CCSS.ELA-LITERACY.RL.4.1
Refer to details and examples in a text when explaining what the text says explicitly and when drawing inferences from the text.

CCSS.ELA-LITERACY.RL.5.1
Quote accurately from a text when explaining what the text says explicitly and when drawing inferences from the text.

CCSS.ELA-LITERACY.RL.6.1
Cite textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

CCSS.ELA-LITERACY.RL.7.1
Cite several pieces of textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

CCSS.ELA-LITERACY.RL.8.1
Cite the textual evidence that most strongly supports an analysis of what the text says explicitly as well as inferences drawn from the text.

CCSS.ELA-LITERACY.RL.9-10.1
Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.

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

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

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

1: Below Mastery 2: Approaching Mastery 3: Proficient

(At grade level)

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

 

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

 

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

 

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

 

Grade 5 Grade 6 Grade 7 Grade 8

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

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

CCSS.MATH.CONTENT.4.NBT.A.1
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

CCSS.MATH.CONTENT.5.NBT.A.1
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

CCSS.MATH.CONTENT.6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.

CCSS.MATH.CONTENT.7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

CCSS.MATH.CONTENT.8.NS.A.1
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

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

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

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

There are other planning documents that show the progression of standards, as you can find here: http://ime.math.arizona.edu/progressions/

And Randall Charles does a great job at summarizing big ideas in mathematics which you can link to the standards: http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigIdeas_NCSM_Spr05v7.pdf

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

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

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

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

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

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

The standards about this theorem exist in 8th grade:

CCSS.MATH.CONTENT.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.
CCSS.MATH.CONTENT.8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
CCSS.MATH.CONTENT.8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
It’s talked about explicitly again here:
CCSS.MATH.CONTENT.HSG.SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
CCSS.MATH.CONTENT.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
It’s the underpinnings for these two standards, but you need to already know the law of cosines is connected to PT to know that:
CCSS.MATH.CONTENT.HSG.SRT.D.10
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
CCSS.MATH.CONTENT.HSG.SRT.D.11
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
So let’s see: if I was an 8th grader, learning about the Pythagorean theorem, would I necessarily be able to use a rubric to understand where I was in my understanding? I’m not sure. Here’s one attempt at making a rubric for it…
1: Below Mastery 2: Approaching Mastery 3: Proficient (At grade level) 4: Mastery (above grade level)
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in mathematical problems in two and dimensions. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two dimensions. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

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

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