## 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.

## 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.

## “Exploring the math shelf #1” – How to pick math books for a meaningful discussion

I often write about how much I enjoy listening to a child making sense of a math problem. But something I find quite fulfilling as well is reading books with my children (any child, actually). No need to say that I am quite excited about our new journey, “Exploring the math shelf”, that takes us weekly to our public library to explore their selection of math books.

Before I start discussing books we have checked out, I thought that sharing our book selection process could be helpful as well. It has indeed been working quite well for us, considering the math discussions that have popped up at our house in the past few weeks.

At some point, I expect, well, I hope, my kids will be naturally attracted to the math section, and pick math books on their own. For now though, I think they count on me to do so :-).

1. I usually do not have any specific math books in mind when I arrive at the library. I like to browse the shelf.
2. Often, I take several books around a math concept my kids have been talking about, such as place value, multiplication or fractions.
3. I try to mix stories (e.g. “Earth Day – Hooray”, by S. J. Murphy and R Andriani, a story illustrating place value) with books specifically detailing a math concept  (e.g. Place Value, by D. A. Adler). That way, we can have an informal math talk while reading a story, and have access to other more detailed sources if my children raise a particular math question.

Once home, we read books when we feel like it. Sometimes at night, sometimes during the day. Unlike with the story books, we rarely read our “math” books from beginning to the end. Rather, we discuss a page or two at a time, such as page 8-11 in Place Value, where the number system is compared to the alphabet. But we go back to the books several times during the week.

One more thing: I do read the math books with my kids. Indeed, you would be surprised by the imprecision, and even inaccuracies you may find in math books. I checked out recently a book about multiplication, and was quite excited to see that the equations included the units. I always encourage my child Rosie, 8,  to do the same when she solves problems (e.g. 2 cats + 3 cats = 5 cats). Indeed, there is no need to wait for a Chemistry course to begin such a helpful habit. Alas, in the book, to illustrate 2 baskets containing each 7 tomatoes , the equation was:

7 tomatoes X 2 baskets = 14 tomatoes.

Somehow, magically, the “basket” unit disappeared (to produce more tomatoes???). Now if the equation had been written as follow, I would have highly recommended the book:

I have to say, though, that once you are aware of possible misconceptions, and discuss such eventualities with your child, it opens a new door to more meaningful math. Rosie is now on a new quest:  “See, Mom?  They say that you have to use the zero mark when you use a ruler. But you don’t have to ! It is easier, but you don’t have to !”. Never too early to become a critical reader.

## It sure is a journey !

When I decided, a couple of years ago, to start this blog, I saw it as a journey, my journey as a parent helping my children with math at home, willing to share all the good stuff I would learn as a M.Ed. student in math elementary education to, hopefully, inspire other parents.

It has, indeed, been an instructive journey. And attending recently a conference dedicated to Cognitively Guided Instruction (CGI) makes me feel like embracing the journey even more. In the past two years, I have enjoyed listening to my children solving problems “a way that makes sense to them” (Carpenter et al, 2014) and meeting the CGI community deeply confirmed my beliefs in such approach. CGI can be complex to describe, but in the context of this blog,  I would define it as a math instruction focused on how children think in math i.e. children’s mathematical thinking: children are invited to explore problems prior to receiving any formal instruction, prior to being introduced to any symbols or procedures. While children make sense of a problem, adults listen. In a CGI classroom, as children share their work, teachers embrace opportunities to build up their math instruction. At home, of course, one may not expect a nurturing classroom discussion. Still, I believe the exploration as such, without time pressure or peer pressure, the “out-loud” thinking is quite valuable. I have opened the door of my house to CGI, and I have enjoyed sharing my experience as a parent on this blog.

I often wonder, though, how I can reach out to more parents. Because it was, of course, my main goal in blogging: helping other parents. I use pen names, which makes it trickier to use social media as a megaphone, and I am more of a “behind-the-curtain” kind of person. So I guess the key will be to get back to more regularity  in my posts. Luckily, I came back from the conference with plethora of new activities to do with my kids, new blogs and resources to explore. So from now on, you can expect to find, once a week, at least one of these posts :

• “Solve and Share”:  I will continue to post word problems to explore across elementary grades but I will include additional children strategies from the literature to complete the experience I have with my own children. Hopefully, these posts will inspire you to welcome CGI at home.
• “How many?”: I am super excited about sharing this activity as I have done it a couple of times with my kids, and they loved it. I will dig further to let you know the genesis of it and explain it in further details when I write our first post but basically, you show the kiddos a picture, and ask them “how many?”. And they can count … whatever they want. They may start with counting items one by one, but the picture can open up to counting by groups, finding arrays, discussing fractions, etc. Here is a picture to help you start thinking about it.

• “Exploring the math shelf”: I naturally add math questions to any book I read to my kids, but I discovered an entire shelf dedicated to math books at our library (I know, it is about time). I started reviewing them, I have to say that some are much better than others. No wonder why kids get easily confused with math ! So each time we go to the library, I will bring a few math books and share my thoughts with you.

Of course, I will continue to post about any relevant matter for parents I read  as a doctoral student. Please, do not hesitate to share your thoughts as well, and raise any questions you may have. Time to fully connect with the math e-community !

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.