## How Many #3? In Beaufort, NC

If you want to start our journey “How Many?” from the beginning, please click here. The goal is to look around  and ask our children:”How many?”. It is up to them to count whatever they want. As always, I hope it helps you see all the counting that can be done around. Search #unitchat on Twitter to find some more !

I took the picture below a few weeks ago and was curious to hear what a child would count. I tried with Rosie, 8, last week, and we had indeed a fun discussion around counting numerals, letters and words. She started counting the numerals written in the arabic numeral system (e.g. 1  7  0  9), and added the numerals written in the roman numeral system (e.g. XII, II).  Then, she counted the letters on the top of the clock (e.g. T   O   W   N , etc), and noticed that the roman numeral system used on the clock was based on …. letters. More letters to count!  We compared the 4 numerals in 1 number (1709) with the 4 letters in 1 word (town). We also talked for a while about the roman numeral system as we read a book mentioning it not so long ago (if you don’t remember, it is here). What is interesting with the clock is that 4 is written as IIII (i.e. 4 ones) . Often, it is written as  IV (i.e. 5 minus 1) similarly to the representation on the clock of the  VI (i.e. I on the right of the V to represent 5 plus 1),  IX (i.e. I on the left of the X to represent 10 minus 1) and the XI (i.e. I on the right of the X to represent 10 plus one). But in this case, if 4 is written as IIII, why 9 is not written as… VIIII ?

It was indeed an interesting picture to discuss, you may want to give it a try.

So:  “How many?”

## Exploring the math shelf #3 – “The Grapes of Math” and other Greg Tang books

“Exploring the math shelf” is a journey that takes us weekly to our public library to explore their selection of math books. Click here to follow it from the beginning. Whether you are a parent, a teacher, someone supporting a child’s math thinking, I hope you find our books review helpful !

This week, we had fun exploring several math books written by Greg Tang.

• The books “Math Fables” and “Math Fables Too” present short stories about 1 to 10 animals gathering in a single group first, and then breaking down into two smaller groups.
• The books “The Grapes of Math”, “Math for All Seasons”  and “Math Potatoes” invite the reader to count items, suggesting strategies to count them other than by Ones (e.g. grouping items in a special way; counting by 5s or 10s, etc).
• The book “The Best of Times” reviews the multiplication facts from 0 to 10 through short riddles.

1. As often with the math books we take at the library, we did not read any of the books from the beginning to the end. Rather, we picked a few pages to discuss at night, or when we had  few minutes to spare here and there. These books have a perfect format to do so, and get a daily dose of math.
2. We spent most of our time with the books “The Grapes of Math” and “Math for All Seasons”, discussing strategies to count. The books give clues leading to one in particular, but we did not read it right away. Rosie came up with her own strategy, and shared it with me first, then, I offered mine, and finally, we reviewed the strategy from the book. It seems a good way to help a child not only build up his/her own mathematical thinking but also make sense of a strategy that may be different from his/hers.
3. Although the books are mostly focused on thinking, a few “tricks” can be found. I decided to skip the ones connected with concepts that Rosie has not fully explored yet. For instance, my hope is that by providing Rosie with plenty of opportunities to explore multiplying by 10, she will notice on her own the particularity of the products. Therefore, telling her now that she can multiply any number by 10, by just adding a 0 at the end seems going backward in our home journey of making sense of math.

I encourage you to check these books out. And if you like them, there are two more (“Math-Terpieces” and “Math Appeal”) you can explore !

## How many #2 – From the captain’s cabin

If you want to start our journey “How Many?” from the beginning, please click here. The goal is to look around  and ask our children :”How many?”. It is up to them to count whatever they want.

We visited an old boat a few weeks ago. I did not ask Rosie, 8, and Tom, 5, “How many?” on the spot, but I took a picture as I was quite curious about what they may decide to count.

So: “How many?”

Tom and Rosie took turns, to count, an easy way to keep them both engaged even if they are at different stage of development in their counting skills.

Tom, counted by Ones: 4 windows, 2 ship wheels, 1 bell, 1 wall, 1 stool.

Initially, Rosie counted by Ones as well: 2 ropes, 20 studs, 1 picture (ah!), noticing details, such as the 5 circles in the middle of the large wheel.

Then, came:

• the array on the stool, how it could be 5 rows and 7 columns of dots. Or 8 columns. Or more.
• the small wheel and its 6 spokes, dividing the wheel into 6 equal parts (i.e.  sixths !)
• the large wheel with its 7 spokes. Wait, there are some hidden ones … there must be 3 more! We ended up with discussing the ten equal parts of the large wheel.

A fun picture to discuss, indeed, and the hidden parts added a lot to the discussion. I hope it helps you see all the counting that can be done around. Search #unitchat on Twitter to find some more !

## Another angle to equity

My graduate studies give me plenty of opportunities to discuss math with my kids. But they also take me on a reflective journey about other crucial components of education, such as diversity and social justice. I often share my thoughts at home as well. Indeed, exploring the concept of equity and social justice is a moving journey that I encourage everyone to embrace. And I don’t think it is ever too early to start.

You may be aware of a cartoon from Craig Froehle (2012), that has been used widely on the web to explain equality vs equity *. It represents 3 people using boxes to try to watch a baseball game behind a fence.

I found it quite helpful to grasp the difference between equity and equality. So I drew a similar picture to my kids. I guess I could have just showed them the initial cartoon, but I knew they would be more engaged listening to the story while I was drawing. Besides, they could imagine whatever they wanted behind the fence.

But then, recently, I found additional pictures, where the attention is no longer focused on the boxes, but on the fence itself.  A new angle to equity.

People can see through the fence. Or even further, the fence is no longer present.

How interesting. How poignant.

Time to go back to my markers and draw a better world.

* an instructive post on the evolution of the initial drawing can be found here.

## “Exploring the math shelf #2” – Building Blocks of Mathematics

“Exploring the math shelf” is a journey that takes us weekly to our public library to explore their selection of math books. Click here to follow it from the beginning. Whether you are a parent, a teacher, someone supporting a child’s math thinking, I hope you find our books review helpful !

Our weekly trip made us discovered a series of books called “Building Blocks of Mathematics” (by Joseph Midthun and Samuel Hiti).

The series comprises six books

• Numbers
• Subtraction
• Multiplication
• Division
• Fractions

I highly recommend them.

1. The books are amusing, cute, strips books. Once we started with the first one, Rosie, 8,  could not wait to read the next ones.
2. We started with Addition, Subtraction, Multiplication and Division, as Numbers was not available initially, but I do not think it matters. Numbers can be read independently.
3. The book Numbers presents a variety of counting systems, and invites children to create their own. I remember having to create my own Base System as a M.Ed. student, it was quite an instructive process, to say the least ! Rosie started by making random symbols for each numeral she would think of. We started discussing about patterns that usually occurs in counting systems.  Her second attempt was quite close to our decimal system, but in her third attempt, a different logic started to appear. Her reflection is far from being completed, but I can see how the book Numbers could indeed lead to a powerful activity around counting.
4. The book Numbers also goes into place value, presenting how some systems have place value while others do no. I have to say that I had never really thought about it. For instance, the counting symbols used by the Egyptians could be written from left to right, or right to left, each symbol keeping the same value no matter its position. With the Arabic numerals, however, the value of each digit depends on its place in the number (e.g. the 5 in 53 has a value of 5 Tens). Interestingly, it seems to me that the Roman numerals are kind of in-between: V has always a value of 5, but IV and VI have different value, depending on the position of the symbol I (4, when I is placed before V, and 6, when I is placed after V). Place value is a concept so often misunderstood, Numbers provides an opportunity to approach it through another angle that would be helpful even in upper elementary grades.
5. In Numbers, there is even a WHOLE page on Zero, a numeral so often forgotten!!!
6. My hope when I pick a book series, is to find some connections between the math concepts presented in each book (e.g. a link between geometry and fractions, or multiplication and repeated additions, etc). This series exceeded my expectations on that front. The character “+” leads the story in Addition, but is also part of Subtraction aside the character “-” and Multiplication along character “x” . In Division, all characters are present (“+”, “-“, “x” and “÷”) illustrating well the relationship between the four operations.
7. The books contain reassuring words (the character “+”, for instance, saying “It never hurts to slow down when you are doing math”, or “you can always use me to check your work” in the book Multiplication).  I have to say that Rosie is not the most confident person around (the apple doesn’t fall far from the tree), and she found it quite comforting to read that when you are stuck in one operation, you can always go back to another one.
8. The characters “+”, “-“, “x” and “÷” discuss different ways to solve problems, using drawings, number lines, equations, etc. A good review of strategies to discuss with your child.
9. Rosie has not been talking about division at school yet. Still, she was fully engaged in the book “Division”, as the concept is clearly presented, and well connected to the other operations she is more familiar with. I decided not to read the whole book “Fractions”, though. It is well written, but I  want Rosie to keep exploring fractions a little further without going to rapidly into their symbolic representation. I look forward to doing our Time 4 Fractions in the Fall for the third time, I may go back to this book once we are done.
10. Cherry on top: the book Cognitively Guided Instruction (Carpenter et al, 2014) is referred as a resource for educators. If you have been following my blog, you know how highly I recommend this approach of instruction :-)

I could still add to the list,we had so much fun reading them. I hope you do to !

Reference:

• Carpenter, T., Fennema, E., Franke, M., Levi, L. and Empson S. (2014). Children’s Mathematics, Second Edition: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. ISBN-13:978-0325052878.

## How many ? # 1 – At the coffee shop

Our journey “How many” comes from the presentations and discussions I had at the Cognitively Guided Instruction conference last month, around counting collections of items (e.g. legos® blocks, buttons, etc) with young children (Carpenter et al., 2016) and how the question “how many?” can lead to math discussions deepening the children’s understanding in number sense in upper grades (Schwerdtfeger & Doto, 2017). Depending on their counting skills, children may explore a collection by counting each item. Others may select a specific feature, such as the pegs of the legos® blocks or the holes of the buttons. Children may count by 1s, 2s, 4s, etc, keeping track of their counting on paper, using cups or bags to group items in Tens, or Hundreds, etc. Children can also count items from their environment, or on a picture (see a post from Christopher Danielson here and Brian Bushart here or search #unitchat on Twitter).  It is endless.

I started the journey informally with Tom, 5, and Rosie, 8, a couple of weeks ago. As we were taking a break at the park, I looked around, and asked them: “How many?”. After an expected “How many what?”, they quickly figured out that they could count whatever they wanted: the cars passing by, the trees, the people, etc. Since then, we have been taking our “how many” breaks regularly. Sometimes for just a few minutes, sometimes for a longer period of time. And we take a picture of what we have been counting. I am going to post theses pictures, with the hope that soon, you will, too, put your “how many?” glasses on. Indeed, I sometimes feel like I am wearing new glasses, looking around for things to count wherever we go…

Here is a first picture, taken at my office our nearby Starbucks®.

“How many?”

What I enjoy the most with the activity is that, although Rosie and Tom are at a different stage of development in their counting skills, they can both be fully engaged in the same discussion.

• Tom started counting items by ones: some of the packages of coffee, the shelves, the straws, etc.
• Rosie, who has been quite curious about  multiplication and arrays for a little while, decided to count the bags of coffee on the top shelf, including the ones hidden. So, 7 rows of 4 packages of coffee… Your child may know that 7 x 4 = 28, Rosie solved it with a repeated addition… it would be 28 packages… That’s when Tom mentioned that there were not 1 shelf but 5 shelves, so Rosie kept adding…

We also discussed why people may want to know how many packages can be held on the shelves. When doing math, we always try to keep in mind the purpose of it…

Your turn to look around!  “How many ??”

References

• Thomas P. CarpenterMegan Loef Franke Nicholas C. Johnson, Angela C. Turrou, Anita A. Wager (2016) Young Children’s Mathematics: Cognitively Guided Instruction in Early Childhood Education. Heinemann: Portsmouth, NH.
• Julie Kern Schwerdtfeger & Darlene Fish Doto (2017) Counting Collections in the Upper Grades (3-5). Cognitively Guided Instruction. 2017 National Conference, Seattle June 26-28.

## Building back up confidence in math throughout Summer

I find it somehow arduous to follow what my daughter, Rosie, 8, learns in math on a daily basis during the school year. But in Summer, I usually am committed to catch up with what I have missed, especially as we usually have quite low key Summer at home.

• Two years ago, I decided to use the Common Core Math Standards (here) to come up with weekly activities to review with Rosie what she had learned in K. We had a fun Summer of learning, but it was quite time-consuming to plan it.
• Last Summer, I tried to take the same path, but could not keep up with the M.Ed. Summer courses I had to take in paralell. Rosie was a happy child in 1st grade, seemed confident in her math skills, so we ended up doing math mostly informally throughout Summer.
• This year, I decided to try something new with my now rising 3rd grader. I had to, as Rosie came home one day, the last week of school, claiming “I know I am not smart, I don’t even know my multiplication facts *”. Sigh**. I have 10+ weeks to build back up her confidence.

So here is what we have been doing:

1. Every day, Rosie explores a word problem “in any way that makes sense to her”, as recommended in Carpenter et al, 2014. We have been using the pool of word problems discussed in one of our previous posts (here).
2. She also works on a daily worksheet from Summer Bridge. I rarely give worksheets to my kids, but I was curious to connect our Summer learning with some of the math activities that Rosie does at School, and see if she needed any kind of reassurance on that front.  There are dozens of books to review Standards throughout Summer, I picked one that does encourage children to explain their reasoning in math. So far, I have liked it.
3. Once a week, we go to the library to pick up books related to math that we read informally throughout the week (click here for more details) .

And so far, it has been working quite well ! You may want to give it a try, it is not too late !

• It is short. Within 15-20 min, Rosie is usually done with her “formal” learning time and has the rest of the day to keep learning… through free play!
• The exploration of word problems has been quite nurturing. She started Summer with trying to remember the procedure she was taught at school, doubting of herself when she could not, to reaching out a new level of confidence, making sense of the problems on her own. As always, I mostly listen, asking questions from time to time to make sure I follow her reasoning.
• Observing Rosie filling up worksheets has been quite instructive as well. Most of the Summer Bridge activities encourage math thinking. Still, a few do not.
• I could see Rosie’s face change the first time she had to solve a dozen addition or subtraction problems in a row (see picture on the right), her eyes begging me to let her skip the few pages providing such a repetitive task. “Let’s just try to make it a little bit more exciting, Rosie. If you had to pick 5, which ones would you pick?”. Now she does not solve them like a machine, she thinks first. “I will do 688+102 because adding 102 is like adding 100 and then 2 more, so I already know it is 790!”.  I understand that practicing a skill develops fluency, but fluency  should not come with… a lack of thinking. At the CGI conference I attended last month, we were shown a video of  a high schooler, enrolled in advanced math courses,  solving 4001 – 3998 … with a standard algorithm (see representation on the left). He was so used to using the procedure that he did not notice that the subtraction could be performed mentally (believe me, it happens to the best of us… See one of our previous posts “When I got swallowed into the symbolic level“). En garde !
• I also noticed how being invited to solve a problem in a little square puts Rosie back in some kind of school tracks, away from freely showing her way of thinking. She did mention though that I should feel lucky, she could have just written 34 or 32, without any units or equations.
• Exploring math book has also boosted our math talks, as discussed here.

We shall see what the rest of Summer may bring, but so far, combining word problems, worksheets, and math books has provided us with a good balance of learning. Indeed, I even found a little fairy waiting for me on the kitchen table last week. A math fairy. She seems rather happy and confident, don’t you think? I just hope she does not fly away at the end of Summer.

* It is by the end of 3rd grade that students are supposed to know their multiplication facts, so obviously, Rosie, you still have plenty of time.

CCSS.MATH.CONTENT.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

** T.J. Zager, the author of Becoming the Math Teacher You Wish You’d Had, shared a similar experience on Twitter last week. Wondering how many 8 year old girls feel that way.

Reference:

• Carpenter, T., Fennema, E., Franke, M., Levi, L. and Empson S. (2014). Children’s Mathematics, Second Edition: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. ISBN-13:978-0325052878.