5 Practices for Effective Teaching!

So when I first learned about Smith and Stein’s Five Practices, it was in the context of facilitating productive discussions in the math classroom. In case you’re not familiar with their book, here’s a link:  5 Practices Book on amazon

Last Spring and this Fall, I co-facilitated a PLT on the 5 practices, and I recognized that they permeate much more of teaching practice now than just when I’m considering a rich task for facilitating a whole class discussion.

The 5 practices are:

  1. Anticipating
  2. Monitoring
  3. Selecting
  4. Sequencing
  5. Connecting

This year, I’ve begun facilitating PDs on CMP3 for NYC teachers. I’ve been noticing how the 5 practices support effectively implementing a Launch-Explore-Summary style lesson: Do the Math before the lesson (i.e. anticipate student mistakes and student strategies), monitor student work during the explore, so you can select students to share during the summary. Sequence the work during the summary so that you are able to draw the connections to the big ideas necessary.

I’ve also been trying out the math instructional routines from Amy Lucenta and Grace Kelemanik. I’ve been noticing the parallels between how I plan for a Contemplate then Calculate or Connecting Representations – the anticipation (including considering how to annotate the student work during class) is the pre-work where we have to do the math in advance. During the routine, we need to monitor student work to select students to share their strategies or their connections, which need to happen in a particular sequence. Then, we’re pushing students to connect to other areas of math and asking them to do a reflection about those connections.

Is it just me who’s noticing the depth to which these 5 practices permeate effective math teaching? Or is there research/articles making these connections explicit beyond just discussion? I’m not sure what’s out there.


7th Grade #CthenC

On December 19th, I lauched my expressions unit in 7th grade through using the Contemplate then Calculate (CthenC) routine I’ve been learning from Amy Lucenta and Elisabeth V at MfA. Based on Amy’s suggestion that it takes three times of doing a routine to get really good at it, I decided to try out three days in a row of contemplate then calculate – and was pleasantly surprised by day 3 when in one class, my students jumped right into writing an expression (skipping “stage 10” altogether!).

We launched on day one with a relatively easy growth pattern of 2n + 2, with the squares arranged in a rectangular array. I flashed this image at my students (for the noticings), and then it disappeared.

CthenC launch

My students noticed the following things:

Class A:

3 sections

named term 1, 2, 3

objects got larger to the right (then the student clarified: the size or number of squares got larger)

Adding 2 from left to right

4 squares, 6 squares, 8 squares

2 columns


Class B:

1st stage had 4 squares

2nd stage had 6 squares

It seemed like it was increasing by 2 squares

Term 1, 2, 3 on top

2 columns consistently


I then told students to work with their partners to figure out how many squares would be in the 11th term. And then I did the hardest part of this routine – tried to scramble around the room, eavesdropping enough to choose students to share their thinking (and confirm with them that they’ll share the ideas I heard), but not long enough to get stuck in any one conversation. I sometimes had to ask students to tell me how they or their partner was seeing it because I walked up when they weren’t talking – but then I felt bad walking away from students who I asked to talk when they weren’t done, but I knew I didn’t need to hear the rest (because I had already chosen someone to share their way of thinking about it, or because they weren’t finished with their thinking).

In class A, I chose two students to share their ideas. The first student saw the columns as one more than the term number each time, so she predicted that term 11 would have two columns that were each 12 long (which she described as 11 + 1), and then she doubled it to get the total.

The other student noticed there were 4 in the first stage, and it was increasing by 2 each time, so to get to term eleven, he said we would’ve added 2 ten times, so he did 2 x 10 and got twenty, which he added to the initial four.

At this point, another student who I hadn’t chosen said she found something similar, but different, and I encouraged her to share out. She said she saw two at the top and then a rectangle of 2 by the term number, so the 11th term would be a 2 x 11 rectangle (with 22) plus the two at the top for a total of 24. I had been searching for this method and hadn’t seen it, so I was excited when she volunteered it.

I then asked the class to consider how each student would’ve thought about the 100th stage and had them share first with their partners. Then I drew it out and wrote the annotations identically to the 11th term. Finally, I pushed my students to generalize and asked them what the “nth” term would look like, explaining that the nth term was an expression for “any term.” (somehow, connecting “any term” to “nth term” has helped my seventh graders in the past make sense of this somewhat confusing notation!). All in all, my students were able to get the three algebraic expressions matching the ideas exactly: 2n + 2, 2(n – 1) + 4, and 2(n + 1). We didn’t discuss the equivalence here.

Class B had 2n + 2 and 2(n + 1) come up, and we had similar discussion around it. No one shared the other expression, and since it wasn’t “necessary” for my main idea, I didn’t push it.

I’m going to pause in describing my CthenC roll-out to go to my final session at MfA about CthenC!

I’m finding that there’s still a bit of a disconnect for me in the theoretical knowledge of how the routine is supposed to be enacted and the practical application. I find that I don’t have enough working memory (or good enough ears/skills of observation) to find all of the students I want to select to share. I have to find better ways of “cheating” – because my students don’t gesture as much as I’d like them to, so I have very little window into their minds except to eavesdrop and I have 27 and 28 kids in my two 7th grade sections!

More on CthenC coming soon!

6th Grade Illustrative Mathematics Unit 4.6 – 4.9

So last year was my first year teaching fraction division, and we did a rushed job of it. It was the last (mini) unit of 6th grade math before the state exam, and I basically taught it as the inverse of multiplication, with some ideas of scaling using ratio tables (from Kara Imm’s number talks on paint cans on her website, numberstrings.com). Many of my students already knew the algorithm from their fifth grade classes, so we didn’t do a great job at teaching that topic. And it was sufficient, but not enough for me. This year, I rearranged the order of our units with my colleague to make the fraction/decimal operations unit come right after our second/third unit, so that after our students have done integers (first), coordinate plane (second), and factors/multiples (third), we would go into fraction/decimal operations. This will allow us to reinforce those calculations within the units on expressions/equations and ratios/proportional reasoning (though, I’m honestly thinking for next year, we might want to do ratios and proportional reasoning FIRST and then fraction and decimal operations – whoops!).

Anyway, since I’ve never taught this topic to the level of depth I wanted to, I decided not to reinvent the wheel, and instead, to look at known resources. While I used CMP3 to launch this unit on fractions and decimals, they go more in-depth into addition and subtraction than we wanted to spend. Plus, their division problems in problem 3 didn’t go into detail about the two different ways of thinking about division (quotative and partitive) and that was something I wanted to try out with my students this year.

I really have been enjoying the lessons from Illustrative Math’s open up resources. I do find that I don’t have enough time to go into quite the level of depth that they do (there’s 17 lessons on just developing the ideas about division of fractions! each of which are anywhere between 40 and 50 minutes depending on optional activities). I’ve had to cut out some of the activities to make it work, but I REALLY really like it. I think next year, I might go straight into this investigation, and not do the other lessons from CMP3 first (it’ll depend on how we rearrange the units and whether we need those ideas of equivalent fractions to be developed in this unit or if they’ll have already been developed in a ratios and proportional reasoning unit first).

On the Friday before break, I had my students do a set of problems with fraction strips. Some of the students used the diagrams, some of the students wrote equations and used the algorithm, and some of the students thought about it in other ways (that I didn’t highlight). What came up was that some students said they were using an algorithm they didn’t really understand and I asked if they could connect it to what the person who used the diagram had done. We ultimately decided that the reciprocal represents the number of the divisor in one whole (which is basically the definition of reciprocal), and we multiply it by “how many wholes we have” (because we were doing whole number divided by a fraction first). I had never realized that interpretation before of fraction division – and I was impressed! I know this is tricky stuff (Liping Ma’s book, the Knowing and Teaching of Elementary Mathematics made me realize how tricky fraction division is – and it’s part of what inspired me to become a math teacher!). But our discussion around these problems was what led them to this conclusion!

This week, I started with trying to get a bit too much accomplished… I wanted to squish ideas from 4 different lessons into just today. In my double period, I managed to go through three handouts and do an exit ticket – and the amount of struggle they had with the exit ticket made me realize I really rushed it too quickly. I’m struggling with the balance of the kids who “already know the algorithm” and the kids who have limited prior experience with division of fractions, and I’m not sure yet how to balance that.

I am also struggling with how to go over some of the topics/skills in the work from Illustrative Math. I’m doing a lot of partner talk (which I normally do), but not as much whole class conversation – because I’m feeling like some of it is tricky to hash out in whole class conversation without looking at their diagrams – but I don’t always find them to be using the diagrams in a way that actually makes sense to solve the problems. So I wonder if I should’ve had more direct-instruction about it? But I think that defeats some of the purpose. I think it went a bit better in the second class I taught today because I knew which things to point out to them in the warm-up (where the diagrams were already half drawn for them) and how to help them connect it to the diagrams in the second handout.

I also made some strategic choices about which items to focus on and which items to cut. I used 4.6 warm-up and created a handout where they were drawing the tape diagrams. I pulled some other questions from the IM curriculum (in the practice sections, etc) and made a two-sided handout. The students got a chance to create the diagrams for a number of problems, some referring explicitly back to the warm-up of “How many groups of 2 1/2 in ten?” and “How many groups of 2 in 7?”

I had the students discuss in partnerships the questions of how they began to draw their diagrams and how they saw the answer in the diagrams (how many groups). I decided not to go over each diagram individually as a whole class, but rather to put up my answer key, share that theirs might not be identical, but they could edit/revise or ask questions about mine. I saw some pencils/erasers moving as I put up my answer key, but no one had any questions they wanted to share with the whole class. I struggle some times – I’m not sure that it was the best way to go over these problems, but no student’s work had stood out to me sufficiently during the work time as being one I wanted to highlight under the document camera, and sometimes, I struggle with what to have students do… Now I wonder if I could’ve had them do a “notice/wonder” discussion with a partner about my answer key instead! (Darn! Next time!)

I also had the students share out their answers to the following question (also from IM, though I’m not sure if it was in the same lesson or not). “Diego said that the answer to the question ‘How many groups of 5/6 are in 1?’ is 6/5 or 1 1/5. Do you agree with his statement? Explain or show reasoning.”

I chose this question because one of my students who was “done early” had this question incorrectly answered with “No, he’s wrong. It should be 1 1/6 because he has 1/6 remaining.” This student didn’t share her thinking with the class and I didn’t want to call her out, so I just tried to play dumb/confused when the student responded that it should be 1 1/5. I asked the class, “Wait, I’m not really sure what she means by it should be 1/5. I thought there was 1/6 left over? Talk with your partner about which one it should be.” I had a few students share out and rephrase/restate each other’s reasoning. Then that first class (with only a single) ended, and I left them to think about making sense of the “remainders.”

My second class did this same problem, and we had a much richer discussion about it. I shared with them “One student in my other class thought it was 1 1/6. What might they have been thinking?” and they were able to articulate that they just took the extra 1/6 and put it on the answer. Then I asked them to think about about what they would say to that student to help them understand why it’s 1/5, and a student used the equation 1/6 x 5 = 5/6 therefore 1/6 / 5/6 = 1/5 (because you want to know “how much of the 5/6 the 1/6 is”). I felt more satisfied with their reasoning, but I’m not sure that every student was there yet, despite the fact that most students gave a thumb vote agreeing with Diego. I struggle sometimes to get whole-class QUICK formative assessment that doesn’t require me to grade their work, but also gives me a view into their thinking. I’m not sure it’s possible – either I need to look at their work, or I don’t get a view into their actual thinking.

Anyway, in that second class, we then went on to 4.7’s fractional batches of ice cream. I skipped over the ropes section (even though I included it in the handout) due to timing, and also because I wanted to stick with the same kind of diagrams and I felt like ice cream would be something my students understood well. I was quite happy with the way I did this section of the lesson, and I think I’ll repeat it similarly with tomorrow’s class.

I gave them the handout and some independent think time. Then we transitioned into “table talk” time where students can continue working alone OR working with partners. When most students had finished Monday, Tuesday, Thursday and Friday (only one table hadn’t), I asked students to turn to their elbow partners and compare their diagrams – discuss how they saw the numbers in the diagrams and how they could use the diagrams to answer each question. The one group where no one had finished all four, I gave my answer key, and I encouraged them to compare their answers with that one – it wasn’t ideal to me, but I didn’t know how else to manage the fact that that group was SO FAR behind everyone else. I don’t think it was an understanding issue, but maybe more of an organization issue? Or a visual-spatial issue of making sense of the diagrams? I’m not sure.

Anyway, I then showed the four diagrams from IM’s “answer key” and I asked students to share a division equation for each day. We got all four equations up on the board (though we almost had one mishap – one student said 9 / 6 = 2/3, but students saw that he had reversed the divisor and dividend, and we corrected it – while that mistake can be valuable, it wasn’t the focus I wanted to go with, so I quickly dealt with it, rather than exploring it more in-depth).

Then I used the alike/different routine and I asked students to discuss how Monday/Tuesday (both had quotients greater than 1) were different from Thursday/Friday (both had quotients less than 1) and how they were alike. We discussed that they all involved dividing by 9 (which made sense to them because 9 cups of milk was required for each batch, and we were really asking the question “how many times does 9 go in?” My students also noticed that none of the dividends were divisible by 9). We also discussed that they were different because of the quotients being greater or less than 1 – and then I asked them how they could predict whether the quotient would be greater or less than 1 without doing the calculation – and they told their partners and then we shared out that they could compare the dividend the the divisor – if the divisor was greater, the quotient would be less than 1, whereas if the dividend was greater, then the quotient would be greater than 1.

At this point, I should’ve made a different instructional decision than I did. I was feeling rushed because I’d only had the students work on 2 of the 4 handouts I’d prepared for our double today, and I was feeling the pressure of time. Unfortunately, I decided to forge ahead with another handout, but it was really pointless to give out. Students barely had any time to finish reading the instructions before I was asking them to share with their partners. Luckily, it was a matching handout, so they still could discuss something, even if they hadn’t finished – but I don’t think it was as productive as it could’ve been. I then showed the answer key, and saw a bunch of kids asking questions – which I didn’t give us time to answer. 😦 Tomorrow, I will go back to this handout, revisit the front, and ask the students to actually work on the back – and then make sense of it together!

The reason I rushed that handout was because I wanted to give an exit ticket. Unfortunately, I don’t think the students were ready for this exit ticket – it made some leaps above where we were in class, and went to the finish line. It’s the cool down from 4.9 (and we had barely even skimmed the surface of 4.8 in class!):

Noah fills a soap dispenser from a big bottle that contains 1/3 liters of liquid soap. That amount of soap will fill 1/2 dispensers. How many liters of soap fit into one dispenser?

Use the diagram below to answer the question. Label all relevant parts of the diagram.

I added in the question of asking my students to write a multiplication and division equation that matched the situation, because I wanted to know how they were thinking about it. I definitely rushed this (there were only two minutes remaining in class when I handed it out) – luckily, I had homeroom next period, and my students were “patiently” waiting in the hallway while my sixth graders finished the exit ticket. I had to encourage some of them to write to me about their confusion if they felt stuck because it became clear that not all of them were ready to answer this question yet.
After reading the exit tickets, I feel a bit of despair and frustration with myself. I don’t think my students are clear AT ALL about how to use these tape diagrams to reason. Out of 32 students, 11 got it correct by clearly using the division algorithm (i.e. they converted both mixed numbers to improper fractions and then multiplied and simplified).
5 students got the right answer, but I’m unclear from their work how – there’s little to no writing on the diagram and not enough reasoning shown elsewhere for me to interpret. They may have computed mentally (I have students with strong working memories!). The way these students labeled the diagrams makes varying degrees of sense – some labeled it mostly correctly, but didn’t use it to answer the question, while others labeled it in an incorrect way that makes sense to me (i.e. labeling three wholes 3, 3, 3, instead of 1, 2, 3 makes sense, but labeling the whole thing 2/3 instead of 2 1/3 doesn’t make sense to me). One of these four students may have actually labeled the diagram BASED on her answer using the algorithm, because she shows the same algorithm as above, but then she labels the diagram 3, 3, 3, 1/2 on the bottom, and 2 1/3 on the top. When it comes to each individual section, she labels the first three WHOLES as 2/3, but then writes a ? in the half container and writes “I’m confused.” for that section – she clearly hasn’t connected the extra 1/3 liters with the extra 1/2 container.
Three students who were sitting at the same table (the table that didn’t finish drawing the diagrams earlier in class!) wrote 7/9 in each of the three sections that are whole (and two kids wrote 7/18 in the half section and one wrote 3.5/9). I’m not sure how they got the 7/9, though, to be honest – any ideas?
The other mistake I saw on two papers (one is a normally strong student, also) was 5/6 – on the other paper I saw it on, I see them write 5/6, 10/6, 15/6, and then later show 2/3 – buy I feel very confused about where those numbers are coming from.
Two students wrote 2 1/3 / 3 1/2 = 1 1/6 or 1 2/12, which I think is because they divided the whole numbers and fraction parts separately? But they didn’t understand how to label the diagram at all!
I had three more students who could write the problem as a division situation, one of whom was also able to label the diagram correctly, but they were confused about how to solve 2 1/3 / 3 1/2 and either didn’t attempt it or got it incorrect (1 1/6 made an appearance again here).
I had one student label 2/3, 2/3, 2/3, 1/3, and even showed it adding up 2/3 >> 1 1/3 >> 2 >> 2 1/3. But then somehow he wrote 3/4 of a liter as his answer. I can only assume that it’s a “typo” because there’s literally nothing else on his paper to suggest 3/4!
I had five students (not including the previous one) write 2/3 in each of the three rectangles and 1/3 in the half rectangle. They were all able to write the equivalent division and multiplication equations as well.
So out of that whole class, only five students seem to have understood the diagrams well enough to actually use them to solve a problem. And ultimately, my goal isn’t even to get them to use diagrams – it’s to make sense of the problems and solve them appropriately (which they can mostly already do using an algorithm!). Now I’m left feeling unsure about tomorrow’s lesson for them as well as what to do with my other class.
I know I should go back tot he handout with the matching (for the other three diagrams we didn’t discuss), and I think I should do the handout I made but didn’t use at all about how much in each group with the goal to answer “how many in 1?” after I do the number talk from numberstrings.com about the paint cans (I think the ratio table is MUCH easier to see than the tape diagrams for this!). I can also do some direct instruction off the board to lead into this skill – I think the diagrams for 10 /5 and 6 / 4 are too easy relative to the “Two cups make two-thirds of a batch. How many cups make 1 batch?” (and it also goes right along with the cups of cheese per pizza that’s in Pamela Weber Harris’ book). I also think we might need to discuss more about how to make sense of the diagrams. I wonder if looking at student work for tomorrow’s warm-up (about today’s exit ticket!) might be helpful (especially if it’s a correct one?) or if there’s something different I should have the students do. It’s clear to me that they’re not USING the diagrams to solve the problems – they’re viewing the diagrams as a separate entity. I don’t know whether this is a case for direct instruction or if there’s a way to get a kid to say something helpful, or if we just need more analysis of the drawings? Comment here with suggestions or hit me up on twitter at @MrKitMath.

Reflecting on the arc of my curriculum

This is only my second year teaching 6th and 7th grade math. I work at a school with a lot of students who score a level 3 or 4 on the state exam; for example, most of the students I teach who struggled in my class last year, still got a 3 or 4! So there are different challenges to teaching these students – I need way more extensions than I’m used to needing. I don’t need to scaffold as much (and in fact, I need to consider carefully whether I’m over-scaffolding and underestimating what they can do on their own).

Additionally, since this is only my second year teaching these two grades, I’m not as familiar with the curriculum: both the content in these two grades and the arc or story of the math. I am still figuring out the big ideas, and how best to build them together. I have a 3 – 5 year “plan” or expectation. Last year, I was just merely considering the topics and skills from a stand-point of making sure that I knew all of the topics AND that I knew the skills myself. Unfortunately, this meant that last year, I didn’t necessarily do my best teaching. Some topics were taught in ways that were more true to my pedagogical beliefs, while others were delivered in a way more congruent with my colleague’s beliefs, because I was using their curriculum to do my instruction.

This year, I’ve started to make changes. I’ve considered how I want to build the arc of the various units I am teaching, and I’m thinking about better problems to use with my students. I’m thinking about how to find the rich tasks for students to engage in, as well as the different instructional models I can leverage for different types of tasks. In the course of participating in my MfA PLTs and other PDs, I’m trying out various instructional routines and best practices – everything from the routines for reasoning from Amy Lucenta and Grace Kelmanik to number strings from Kara Imm and Pamela Weber Harris. I tried out an “important stuff, neat stuff, tough stuff,” style packet from PCMI. I have been using some of the content from CMP3, as well as some of the content from Openmiddle.com, openupresources from Illustrative Mathematics. I’ve continued to use some of the resources from my colleagues (especially in the first seventh grade unit on Probability, since it’s drawn from Engage NY in a very specific manner – though I confess that’s currently my least favorite unit because I feel the least ownership and understanding and enjoyment of it).

I am still tweaking the arcs of my curriculum, and considering how and when to cover the various content areas. I’m considering how to pull the various resources together to create a coherent set of units and to address the major skills and avenues for thinking. I’m considering how to build a mathematical community of learners and how to strengthen my students’ mathematical curiosity. I’m enjoying this work, but I do sometimes have to take the “long vision” – I’m not going to perfect my curriculum overnight. I am still making sense of the big ideas and mapping the topics that I have to teach to the fundamental layers of mathematics.

This week in particular, I spent a lot of time using the Illustrative Mathematics 6th grade unit 4 on division of fractions and thinking about which parts were necessary for my students and my classroom, and which things were unfeasible. I’m excited to hear the conversations my students are having, and the ways in which they’re developing their understanding. Earlier today, I had one student confess to the whole class (after multiplying by the reciprocal), “I know to do that, but I don’t know why.” We talked about making sense of the algorithm, and connected it to the diagram another student had drawn – suddenly, it became clear that the reciprocal was the number of that fraction in one whole – and multiplying it by how many wholes you had would get you the total number of that fraction in the dividend. We haven’t done every experience in the Illustrative Mathematics curriculum (and we’re going out of order because my school has a particular arc to our units), but it was such a rich conversation. I’ll write more about it in a separate post. My students have been making sense of models.

Meanwhile, in 7th grade, I’ve been using the routines for reasoning to develop my students’ understanding of how to write equivalent expressions. We did three Contemplate then Calculates to develop my students’ ability to chunk diagrams and see how to find the tenth, hundredth, and nth stages. Today, we launched a new routine: Connecting Representations. I was surprised by the results of it – and again, I’ll make a separate post about this.

The other thing I’ve been doing is collecting more and more student work and analyzing their thinking. In teaching a rational number operations unit in 7th grade, I made a close analysis of the kinds of mistakes the students were making on the quizzes. This will help me both this year, in teaching some of those same skills to my 6th graders as well as going back and doing targeted instruction for students who still made certain mistakes. It will help me even more next year, in planning specific activities to help my students unpack certain common misconceptions and develop a deeper understanding.

Finally, I’ve been thinking about ways to engage my students. I don’t want any of my students to be bored, I want to manage the range of learners in my classroom, and I want to develop a mathematical community more. I have begun to notice that there are fewer and fewer hands going up in my classes (especially in the 7th grade!), and I constantly hear from the same students. I am seeking new routines to change up who the talkers are. I don’t want to do random cold-calling, but I think in January, I might ask students to fill out a survey/reflection about their class participation and to start tracking which students are raising their hands and/or being called on/sharing.

The other thing I want to work on is the small group and partner conversations being more equitable. I use visibly random groupings (students get a card each day to find their new seat), but I’ve begun to notice that when I approach a random group and ask them “What have you discussed?” during a turn-and-talk, it is almost always a boy who jumps in to answer first. And if two students both start talking simultaneously, the boy often talks over the girl to continue and the girl usually backs off to allow the boy to finish his idea. I’ve started naming students and making eye contact with the girls to be conscious of hearing more of the girls in my math class. Probably another post on that at some point.

Sometimes, I have trouble wrapping up my thoughts in a post like this, because I don’t have the answer to the questions I pose. Continue the conversation with me on twitter at @MrKitMath.

Fractions, Decimals & Mixed Numbers, Oh My! #MTBoS HELP!

When I taught 8th grade math, my primary content focus was to master linear relationships. I learned everything I could about slope and y-intercept. I expanded into systems of equations. I discovered a variety of techniques to use to solve problems and many ways of teaching students how to do it. I learned multiple models and I got satisfied with some of them.

When I switched to 6th and 7th grade last year, I knew integer operations were something I was going to need to learn how to teach well. I did some research, and I settled into using a combination of the patterns for multiplying and dividing (based on CMP3’s accentuate the negative), and using the hot air balloon game with addition and subtraction (and even with absolute value and comparing numbers to some degree). I think I do a pretty decent job of teaching it (though obviously, it’s not perfect and some students struggle with some of the concepts, even by the end – especially with subtraction!).

Now I’m recognizing the next content area I need to focus on mastering: fraction/decimal/mixed number operations. Although the focus in 6th grade is on fraction division and on all decimal operations, I also think it’s vital for the students to master the other operations as well – and in my seventh grade class, we are focusing on the operations as well, especially with mixed numbers and combining decimals with mixed numbers (and including negatives!). So I need to find classes, resources, etc. to deal with those ideas next.

What are your favorite resources for multiplying mixed numbers? Dividing fractions and mixed numbers? For addition and subtraction with mixed numbers (especially negatives!). For dealing with decimals in general?

#MfAMT2 Become the Subject (A Pedagogical Use of Self)

Tonight, I had the honor of giving a TED-style talk at the Math for America annual event, Master Teachers on Teaching (Affectionately known as MT-squared). This year’s theme was Truth Matters: Trust, Lies, and Logic in the STEM classroom. This theme really inspired me to share my evolution as a teacher, so I wrote a proposal, and was one of eight teachers accepted to give a speech. I remember bringing my rough draft to the first session of our mini-course to prepare for giving the speech, and feeling like I had finished my speech and just needed to shorten it. After getting some feedback, I realized that I had somehow written 5 different speeches overlapping each other and I needed to pull out one speech to deliver. And I only had ten minutes (my original draft was more like a 25 minute speech!).

I ultimately selected my notion of a pedagogical use of self to focus on, and I wrote this speech. I’m sharing the text of what was written here as well, but it’s not a perfect transcript. MfA asked us to memorize our speeches, and while I did a pretty good job, I had a bit of stage fright in the beginning, and literally forgot my lines! I had to check my script to find my place. Luckily, once I told my story about Bubbe’s bagels, I got more confident, and I delivered the rest of my speech with fidelity. I even got comfortable enough to ad-lib a comment “Thanks Kid,” – which for those of you who know how awkward I can be, especially with improv and delivering speeches, you can understand why this was such an achievement.

My next steps with the idea of pedagogical use of self is that Kara Imm and I are collaborating on writing an article for the NCTM middle school teacher journal. Hopefully, they’ll accept our article!

The text of my speech (as written):

Become the Subject

Kit Golan, Master Teacher

Good evening. I’m delighted to talk with you tonight. First, let me share a dirty little secret: math class is not as fun as science class! Controversial, I know. I realized this as a first-year teacher when I taught both. The problem is I’m a math teacher, and passionate about it!

Those of you who are science teachers likely enjoy designing classroom experiments that facilitate discussions by unpacking student observations. I remember one such experiment, my first-year teaching, when I put food coloring in cups of cold and hot water.   My 6th graders watched, fascinated, as the color spread much more quickly in the hot water.  In an animated discussion, we made connections to brewing tea in hot water, and ultimately spoke about the impact of temperature on the speed of molecules. My students were excited and talked enthusiastically about the experiment.

Later that day, I taught those same students a math lesson. They showed much less interest in adding fractions with unlike denominators. Class became a battle: I spent all my energy trying to get students to do the math, while they spent all their energy trying to distract me and avoid doing math. “What are you doing this weekend, Mr. G?” they asked. “Do you have a girlfriend?” “Do you have any kids?” When asked a personal question, I stonewalled with, “That’s off topic” or “That’s irrelevant.” I thought I was keeping them on task. But what I didn’t realize was that some students were genuinely curious about who I was and wanted to connect with me, and I was rebuffing their efforts!

This leads me to the challenge we face: “How do we provide students with classroom experiences that promote rich discussions and engage their curiosity in math class?” My principal advised, “Just make your lessons more engaging.” “But how?” I asked. I hadn’t learned how to do that in grad school! Meanwhile, my students seemed overly curious about me and uninterested in the math. Then, during my second year of teaching, I had an epiphany and realized that I could use one problem to solve the other!

The breakthrough came after I had delivered a boring lesson on converting Celsius to Fahrenheit. My coach observed that while my students were plugging numbers into the formulas, they had no motivation—it was sheer drudgery. She suggested an alternative: What if I had launched the day’s lesson with a story? Suppose I said I’d gone to Canada, checked the weather before going outdoors and it was 20 degrees. I bundled up tight, expecting it to be bitterly cold, but when I got outside, I was sweating. What do you think happened? This would pique my students’ interest and elicit from them that I’d interpreted a temperature in Celsius as Fahrenheit. Now we had a REASON to learn how to convert temperatures – so we wouldn’t make the same mistake Mr. G made on his Canadian trip.

Using storytelling to hook students and help them learn math was a brand-new idea for me. Immediately, I found it highly effective. The first time I tried this, the kids were more attentive and enthusiastic than I had ever seen them. And so, I became a storytelling teacher. The purpose of my stories wasn’t to tell the literal truth about my experiences, but to cultivate rapport with my students and develop a reason for the mathematics. I leveraged student curiosity about me to engage them before they even realized we were solving math problems. Gradually, I evolved from fabricating stories to turning actual incidents in my life into math problems.

For example, last year, I launched a problem in class like this. “How many of you like bagels?” [Encourage audience to raise their hands]. Me too! I love bagels. We’re lucky we live in NYC, because we have the world’s best bagels. Sadly, my Bubbe doesn’t live in NYC; she lives in Connecticut, where they don’t have great bagels. So like any good grandson, I brought her New York bagels whenever I visited. One time I had to visit on a Monday, when my local bagel store is closed, so I couldn’t bring her any bagels! Let me tell you, my Bubbe never let me hear the end of that! Every subsequent visit, she would ask, “Did you bring the bagels this time?” “Can you believe he forgot the bagels?” So to make sure it would never happen again, I did a little bit of research and I found two other good bagel stores in my neighborhood: Bob’s Bagels and Tom’s Bagels. And both are open on Monday! Can you all help me figure out where I would get the better deal on bagels for my Bubbe?”

My students really got on board with this problem. As I monitored student discussion, I heard lots of conversations about their favorite bagel stores. Students analyzed each other’s methods of determining which store offered the better deal. They were seeing for themselves how mathematics is a powerful tool to solve problems in daily life.

At the end of the year, I asked my students to write a letter with any advice on how to be a better teacher and how they would like me to remember them. One student wrote, “Now, as for any advice, I liked how you would often turn your stories into math problems, it makes math more fun… I want you to remember me as… ‘the kid who reminded you that you forgot to bring your grandma bagels that one time.’” That story had stuck with him through subsequent units, and he recalled it as THE thing he wanted me to remember about him. Clearly, my stories resonated with him.

In building relationships with my students, I’m drawing on what I know about relationships in general. In psychotherapy, there is a term for using yourself to create empathy and a relationship with your patients: a therapeutic use of self. I propose that teaching needs to coin its own term: a pedagogical use of self. A pedagogical use of self is when you strategically embed yourself into the curriculum in stories that will captivate students and cultivate a community of mathematicians—or scientists—in your classroom. Such sharing about yourself will strengthen your relationships with your students. Your curriculum will come to life, your students will get to know you and you’ll draw them into your subject matter.

By using my life pedagogically, I model what mathematics can do for anyone and show students how math can be a tool to help us make sense of, explain, and evaluate our own lives. In eight years of teaching, I’ve evolved from following the lie “Don’t smile until Christmas” and worrying about staying on task, to spending the whole first day of school getting to know who my students are and introducing myself to them – everything from the dog I have to the absurd number of board games I own. By sharing my truths with the students, I build trust and inspire them to share their own truths with me. As a result, I am currently experiencing powerful, vibrant relationships with students and enjoying seeing some of them develop a passion for math.

I hope my evolution will inspire you to embrace a pedagogical use of self as a valuable tool in your own classrooms. Ask yourself, “What are your Bubbe’s bagels stories?” Thank you.

My speech can be viewed on twitter and facebook, by going to Math for America’s page and watching the live stream. I closed out the first half of the program, before intermission. On the twitter version, I get up to talk at 53 minutes.

Time Well Spent #MTBoS

I struggle with time management, especially when I’m lesson planning. I type this even as I should be lesson planning for tomorrow or grading! But I had this realization just now, as I was sorting through different ideas about how to teach tomorrow’s lesson. I was trying to create a lesson about divisibility rules (that could spring somewhat organically from this week’s lessons about GCF and LCM), and I realized that I could use the puzzle I have to launch a NEED to have divisibility rules. Then I found Prime Climb, and I realized that I could further using that to introduce the idea of prime factorization – which some of my students already know about, but this could deepen their understanding and connect the two.

So while I spent more than 30 minutes just thinking about the tweets I wrote a few weeks ago and reviewing the things people shared with me, I feel like I’ve made a much deeper investigation that starts with a question, gives motivation for discovering the rules, and connects them to the larger picture of the unit. All good things.

I guess my conclusion is that GOOD lesson planning requires more time! 😦 Too bad time is so finite!