## Sunshine shenanigans

1. My inbox is where chain letters go to die. I never forward them on.  I do not “Like” photographs that have charming children (or puppies) holding signs asking for 1 million likes. I will not apologize for this.
2. I make a mean beer can chicken.
3. The wings on these chickens pretty much never make it to the table. Seriously, have you ever eaten the wings off a well made beer can chicken right off the grill? (Sorry, vegetarians—much love, but this is about me.)
4. My family never did nicknames growing up (OK, I can think of two exceptions—my mom called my sister Pumpkin and everyone called me Keefer). My wife’s family is rich with them. Our married life has taken on her family’s tradition. Among the nicknames in our house are these (very small sampling—you can have fun at home guessing which applies to whom): Dog, bird, pigeon, Boo, hompish, EP, LP, hound, rabitsu.
5. If there is no urinal available, I prefer to sit.
6. If your band has an accordion, I will gladly come hear you play. I cannot explain this.
7. I am a daily newspaper reader. Paper copy. Electronic is no substitute.
8. I am a huge introvert. Not shy, but being among lots of people wears me out. As a consequence, I relish my quiet down time at home.
9. You will need to tear my Mac from my cold dead hands. There is no other technology about which I feel so passionately. My laptop and I are a team. We need each other to get stuff done.
10. I am a much better person for having met my wife. Rachel is in very important ways my polar opposite and I have learned a lot from her about empathy and humanity. We also, of course, share essential core values.
11. I make pickles. House specialty is a half-sour dill, which is fermented for about a week and doesn’t keep more than another week or two. The result is that they are seasonal. And seriously delicious.

I got nominated three times for this silliness. Fortunately, only two of these involved questions. I am compiling the master list of 22. Copy-and-paste, and hope there is some overlap. Here goes…

1. Why do you teach?
I am fascinated with how people’s minds work. Trying to think like someone else—to see the world from their perspective—is endlessly interesting to me. Mathematical thinking is where I am most skilled at this. Teaching exercises this skill.
2. If you didn’t teach, what would you do for a living instead?
I don’t know. I could probably be happy somewhere in the food industry. It would have to be somewhere that allowed for creativity and problem solving.
3. Money being no obstacle, where would you like to visit? Why?
I want to go back to Japan. Rachel and I visited in 2002 and I found Tokyo amazing.
4. Kids always ask who your favorite student is.  Describe the characteristics of yours
I love the ones who are trying their best to grow. The ones who are satisfied with their present selves frustrate me. The growth they seek does not need to be mathematical, but it needs to be visible in our teacher-student relationship somewhere.
5. What is your favorite board game and why?
Chess. I am not that good and I do not have much opportunity to play. But the complexity that arises from a simple set of rules is beautiful. As is the fact that the game is about ideas. If you do not play chess, this probably makes no sense to you. Sorry.
6. What is the most frustrating component of education right now?
That U.S. teachers are increasingly put in corners where they feel (rightly or wrongly) that theirs is not a creative profession, and that they have limited autonomy to make important classroom and curricular decisions.
7. Would you rather buy a Mac or a PC?
See fact 9 above.
8. What is your favorite book?
Can’t pick one favorite of all time. Recently I read Children’s Minds by Margaret Donaldson which was amazing for where I am in my own work and thinking right now. That’s my favorite recent read.
9. If you had to choose blogging with no way to share it (ex. via twitter) or tweeting with no way to elaborate (ex. via a blog), which would you choose?
Blogging with no way to share. For sure. I blogged for two years before finding my math nerd friends on Twitter. I have too much to say, and I work out my ideas by saying it. I have to write.
10. Who is your hero?  Why?
For me, hero suggests a lack of faults. We are all too complex for that which is why comic books exist. I wrote about important mentors for me in life and work a couple years back. Those people are still tops.
The moments of engagement with students’ ideas. Those moments when ideas are on public view and the classroom community is considering them, changing them and adopting them. I live for those moments. They are more frequent for me the longer I teach and that feels like a reasonable measure of success.
12. If you had to pick one area/concept of math that is your “jam”, what would it be?
Fractions. Next question?
13. To quote Rodney (Chris Rock) from Dr. Doolittle, “You can’t save them all, Hasselhoff.” True, but there’s at least one student that sticks out in my mind that I feel I failed. Do you have one?
Yes. Joe. My last year in the classroom. He needed more weaning from the teacher as answer key than I gave him, and this led to him shutting down too often. I needed a more nuanced approach with him and I didn’t realize it until too late.
14. Twenty years from now, what’s something kids will probably remember about you (phrase, moment, habit, characteristic, etc.)?
15. I nominated you because I think you’re great, but I know we are all our own worst critics. What’s something that you’d like to “fix” about yourself in your current job?
Timeliness in responding to student written work. I am working on this. Part of it is straight-up self-discipline. Another part is being proactive about when and what I collect, and about what kinds of feedback I promise. I am working on both parts.
16. Name a movie title that describes you and why.
Definitely not Stand and Deliver. I will choose from the movies currently playing at my local multiplex (and I will avoid the easy Frozen!)…
…hmmm….
American Hustle. That was fun. There were some funny options. Why American Hustle? I am always hustling in the classroom. Coaxing, marketing, anything to get those minds open.
17. I love TMC because at night I can hang out with my favorite tweeps over a beer or two (or eight). Which tweep would you love to have a conversation with over a beverage?
I have been blessed with opportunities to do this many times in the last couple of years. I’ll pick a couple who I haven’t had the chance with yet. I would like to brainstorm Would You Rathers with John Stevens. I would love to talk fraction learning with Nicora Placa. I have shared a beverage, but not a real conversation with Fawn Nguyen; that needs to change. I haven’t met Andrew Stadel yet. This also needs to change. Jason Buell, Jose Vilson…all of these are people I would love to talk with over coffee or beer, and have not yet.
Also, as a heads up: I accept all such invitations I can make time for. Invariably these invitations lead to me being described as “not as strange as I expected”. I am OK with this.
18. If you couldn’t teach your specific subject, what else would you teach?
Kindergarten. Can I still teach math as part of the day?
19. Everybody has a song they car dance/jam out to. What’s yours?
Dig the rhythms and the horns here. For me the “lovers” are my work commitments, which frequently become too numerous.
20. TMC13 enlightened me on karaoke night. A few people completely blew my mind (I’m lookin’ at you, Pershan). Who would you love to see karaoke at TMC14 and why?
I want an encore from Karim Ani and Eli Luberoff.
21. What’s one thing (item, app, software, etc.) that you love so much that you can’t imagine doing your job without it?
I mentioned my MacBook earlier, right? Seriously. When I moved from MSU, Mankato to Normandale, I asked for a MacBook and was told no. So I bought my own.
22. If you could job shadow one tweep for a week, who would it be and why?
Sadie Estrella. I need to get out of this cold.

Now. You remember how I said chain letters go to my Inbox to die? I can’t do the 11 nominations and 11 more questions.

I will thank Ms. Hedgepeth, Mr. Stevens and Ms. First (sorry, I tried to find a name on your blog) for the nominations. Thank you!

## Ginger ale (also abbreviated list of Standards for Mathematical Practice)

We have some of these mini cans of ginger ale in the house this week. I am not sure where they came from; only that my wife bought them. Normally we only have sparkling water around, not pop (nor soda, nor…)

So I’m looking at the can instead of grading like I should be and I notice the “25% fewer calories than regular ginger ales” claim.

And I think what any skeptical consumer ought to think. Sure fewer calories in the mini can. Duh.

Then I see this:

They have controlled for the size of the can. Nice. This one has 60 calories per 7.5 fl. oz. Regular ginger ales have 90 calories per 7.5 fl. oz.

I am briefly satisfied. And impressed.

But wait! 60 is 25% less than 90? ARGH!

Two possible explanations:

1. 25% means at least 25%, and Seagram’s chose this nice simple number over the more complicated $33\frac{1}{3}%$.
2. It really is exactly 25%. But we know that calorie counts are rounded to the nearest 10 calories.

This second explanation leads to a sort of lovely task. How can we characterize the set of possible calorie counts for 7.5 fl. oz. of Seagram’s and of regular ginger ale so that, (a) the counts round to 60 and 90, and (b) one number is exactly 25% less than the other?

Extra credit: Which standards for mathematical practice are you using as you solve?

Double extra credit: Which of my abbreviated list of standards for mathematical practice (see below) are you using as you solve? And which was I using as I gazed at my can of ginger ale?

Prof. Triangleman’s Abbreviated List of Standards for Mathematical Practice.

PTALSMP 2: Play. See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

PLALSMP 3: Argue. Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

## Webinar Thursday evening!

I was invited by Discovery recently to do a webinar in their Siemens STEM Academy Connect series.

I say yes to pretty much everything that doesn’t cost me money, so I said yes.

It takes place Thursday, Dec. 5 at 7:00 p.m. EST.

This will be version 3 of “Standards for Mathematical Practice: They’re Not Just for Students Anymore!” Version 1 I did for a Global Math Department meeting. Version 2 I did at the NCTM Regional in Louisville. Version 3 is substantially revised from those first two.

Among other things, we will discuss what this photograph has to do with mathematical practices.

There is a handout which I would encourage participants to spend time with beforehand.

It is free.

I have done my best to plan a maximally interactive hour, given the webinar format.

## Summary and wrap up [TDI 7]

It’s finals week here at the Triangleman Decimal Institute.

Let’s review for the final exam, shall we?

Is that decimal point in the right place? How can we know?

Week 1 (Sept. 30): Decimals before fractions?

In this week, we considered the question, Should we treat decimals more like whole numbers or more like fractions? Associated with that question, we strived to make explicit the ways in which decimals are like whole numbers and the ways in which they are like fractions. We covered a surprising amount of ground on these questions.

Week 2 (Oct. 7): Money and decimals.

Our consideration of money and decimals led us to think very, very hard about units. In the case of money, it seems that we reached consensus that the place value aspects of the American monetary notation system are subordinate in many people’s experience to the different units aspects. That is, we are more likely to think of \$3.50 as three dollars and fifty cents—two different units—than as three and fifty one-hundredths dollars.

While both ideas are correct, this conceptual difference has fairly strong explanatory power. It helps us understand a lot of the errors we see in classrooms and in the larger world.

Week 3 (Oct. 14): Children’s experiences with partitioning.

In this week, we explored experiences students have with cutting things into pieces, which is the real-world knowledge children can bring to classrooms when they study fractions and decimals.

Week 4 (Oct. 21): Interlude on the slicing of pizzas.

Dozens of math teachers on the problem led to our documenting 6, 8 and 10 slice pizzas directly (where a slice is interpreted to be the result of complete and equal partitioning of a pizza). We have strong claims that other numbers of slices exist, but no photographic evidence.

Week 5 (Oct. 28): Grouping is different from partitioning.

We argued this week about whether moving to the right in the decimal place value system is really a simple extension of moving to the left. It seems assumptions are everything here. Maria and I brought differing assumptions to this question, and this led to some very interesting and spirited debate.

Week 6 (Nov. 4): Decimals and curriculum (Common Core).

In week 6, we thought about the relationship between the ideas we had been working on and standards documents.

This brings us to week 7.

Your final exam consists of one question and one task. The question is this:

How can you show the world what you have learned these last several weeks?

Do it.

And let me know what you do, OK? I need data for my funders.

## Decimals and curriculum (Common Core) [TDI 6]

The Decimal Institute is winding down. This week, I have a short post outlining the relationship between our discussion these past weeks and the Common Core State Standards (with links). Then next week we will wrap up with a summary of what I have learned and an invitation to participants to share their own learning.

The Common Core State Standards build decimals from the intersection of fraction and place value knowledge. Fractions are studied at third grade and fourth grade before decimals are introduced in fourth:

Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

One of the issues we have been wrestling with in the Institute has been how much decimals are like whole numbers and how much they are like fractions. In light of this conversation, I found the following statements about comparisons interesting.

• CCSS.Math.Content.1.NBT.B.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
• CCSS.Math.Content.2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
• CCSS.Math.Content.4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

These all refer to comparisons of whole numbers—at grades 1, 2 and 4. Comparisons of decimals appear at grades 4 and 5. For example:

• CCSS.Math.Content.4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. [emphasis added]

The phrase, Recognize that comparisons are valid only when the two decimals refer to the same whole, struck me as odd. If I am comparing 0.21 to 0.5, I need to make the whole clear, but if I compare 21 to 5, I do not?

This seems to be an overcommitment to decimals being like fractions rather than like whole numbers. Or not enough of a commitment to the ambiguity of whole numbers.

In any case, the treatment of decimals in the Common Core State Standards is probably one of the major challenges for U.S. elementary teachers, who may be accustomed to curriculum materials that emphasize the place value similarities of decimals to whole numbers rather than the partitioning similarities to fractions.

I will provide some examples of pre-Common Core U.S. curriculum in the Canvas discussion to support this claim. Join us over there, won’t you?

Non-U.S. teachers, please share with us your observations about how these standards relate to curricular progressions you are using. An international perspective will be quite useful to all of us.

And please start thinking about what you can do in the coming weeks to share/demonstrate/document/extend your learning from our time together. Consider it your tuition to the Institute.

## Grouping is different from partitioning [TDI 5]

After last week’s pizza-slicing interlude, we are back on task for the closing half of the Decimal Institute.

This week, I want to invite discussion of the question, How much are decimals like whole numbers?

In case you are this far into things and cannot guess my answer (and in case you haven’t read this week’s title!), I offer the following clue.

From a purely abstract and logical perspective, decimals are exactly like whole numbers. No matter what place you are considering, the place to the left is worth 10 times as much, and the place to the right is worth $\frac{1}{10}$ as much.

But there are many important ideas that this logical analysis ignores. And people do not always find abstract logical arguments compelling. So we’ll dig deeper than that.

I have four major ideas for us to consider. You, class, will surely have more.

1. Grouping groups is different from grouping units. Thanheiser (2009) demonstrated that some preservice elementary teachers could work competently with two-digit numbers yet make important errors with three-digit numbers. These teachers could explain the grouping inherent in writing a number like 23, but did not extend this reasoning to numbers such as 235. If decimals are really just like whole numbers, we should expect that all whole numbers are the same for learners. Thanheiser has demonstrated that they are not.

2a. Grouping patterns and partitioning patterns are often mismatched. The metric system was established by the scientific community for ease of working with our base-10 numeration system. It was developed intentionally at a moment in time when correspondences between numeration and measurement were of increasing importance.

Other measurement systems probably reflect the informal and natural ways people have of working with measurement. The Imperial system, for instance, is probably based on how people naturally view quantities.

In that case, consider the inch. Inches are grouped in twelves. They are partitioned in twos and powers of two.

The teaspoon is grouped in threes (making tablespoons) and partitioned in twos and fours.

Cups? Those are partitioned in twos, threes and fours. But they are grouped only in twos.

Time and again, the size of the grouping is not related to the number of partitions. Perhaps this is because partitioning and grouping are not closely related processes in people’s minds.

2b. This is borne out in my own work with preservice teachers. Go read my post titled, Measurement explored for full details. My experience in having students develop length-measurement systems includes these observations:

1. Students nearly always partition in 4ths, 8ths and 16ths.
2. Students almost never partition into 10ths.
3. Students may group in threes or sixes, but they never ever partition this way.
4. Students rarely think to group the same way they partition. That is, if they made 8ths, they might very well group in sixes. The convenience that would be afforded by consistency does not tend to occur to them in advance.

The comments on that post are thought-provoking and we should feel free to pick up threads of those comments in this week’s discussion.

3. Place value understanding does not seem to cross the decimal point easily. I do alternate place value work with my preservice teachers. Bear with me on this if you’re not familiar. In a base-5 system, we count 1, 2, 3, 4, 10. We make groups of this many: ***** instead of this many: **********; the latter is what underlies our usual base-10 system.

This means we write $10_{five}$ for our usual five and $100_{five}$ for our usual twenty-five. After mastering grouping with fives instead of tens, we move to partitioning. If decimals are just like whole numbers, this should present no difficulty.

But it presents tremendous difficulty. Even my strongest students have a common struggle, which is this: They view the whole and the part of a decimal number separately and treat them equivalently.

Here is what this means. Consider the base-10 number 20.20. This is “twenty and twenty hundredths”. My students tend to correctly interpret whole number part of this. Twenty is four groups of five so they write $40_{five}$. But then they do the same thing with the decimal part, writing $.40_{five}$, so that $20.20_{ten}=40.40_{five}$.

But this is not right. The decimal part represents 20 hundredths. But if we have changed bases, then the values of the decimal places change too. The first place is fifths; the second is twenty-fifths; and so on.

Through the use of grids and activities paralleling those from the Rational Number Project (Cramer, et al., 2009), they come to understand that $20.20_{ten}=40.1_{five}$

The underlying difficulty seems to be that…

4. The unit changes when we add digits to the right of the decimal point. When you read whole numbers aloud, the unit is always the same—one. Thirty-two means thirty-two ones. 562 means 562 ones. Yes, the 6 has a value, and this value changes depending on its place. But no matter the number of digits, the number counts ones.

This is not true with decimals. 0.32 means thirty-two hundredths. 0.562 means 562 thousandths. Thousandths are different units from hundredths. The unit changes to the right of the decimal point in way that it does not for whole numbers.

To summarize, our question this week is: How much are decimals like whole numbers? My answer is that they are not very much alike at all. I outlined four reasons: (1) Even whole number place value is more challenging than logic suggests, (2) Our experiences with grouping and with partitioning tend not to parallel each other, (3) We tend to think of whole number parts and decimal parts as separate things, and (4) The units we count are different to the right of the decimal point, depending on how many digits there are.

How say you, class?

## References

Cramer, K.A., Monson, D.S., Wyberg, T., Leavitt, S. & Whitney, S.B. (2009). Models for initial decimal ideas. Teaching children mathematics, 16, 2, 106—117.

Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers, Journal for research in mathematics education, 40, (3), 251–281.

## Presentations on the horizon

I have been careening from one class to the next, working on projects in between. Doing a tremendous amount of writing. Teaching a hastily organized online course on decimals. You know, as one does.

And then last night it occurred to me that I really should look more than a day or so ahead.

So I did.

HOLY CRAP!

I have some work to do. This will be fun, though. If you find yourself in any of the following locations at the right time, do stop by and say ‘Hi’.

Louisville, KY. NCTM Midwest Regional. November 7, 2013. Session 52: Standards for Mathematical Practice: They’re Not Just for Students Anymore!

Oconomowoc, WI. Wisconsin Math Council “Math Proficiency for Every Student” Conference. November 15, 2013. Two sessions:
1. (Almost) Everything Secondary Teachers Need to Know about Elementary Mathematics.
2. Practicing the Five Practices.

Wausau, WI. Wisconsin Math Council “Math Proficiency for Every Student” Conference. December 13, 2013. (Almost) Everything Secondary Teachers Need to Know about Elementary Mathematics.

Augusta, ME. ATOMIM Spring Conference. April 3 and 4, 2014. Various sessions.

Duluth, MN. Minnesota Council of Teachers of Mathematics Spring Conference. May 2 and 3, 2014. Various sessions.