Tag Archives: Education

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 !

MathFable

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.

Few thoughts about our readings :

  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 !

 

 


Building up a bridge between home and school

I often refer to being on a journey as the author of this blog. I see myself as a lifelong learner, trying to connect my experiences as a parent, as a teacher, as a graduate student in math education. I feel like I am walking in the woods, enjoying the hike, wondering what the next curve may bring.  I pass the curve, and keep on going. Sometimes, I feel like I am getting lost. Sometimes, I reach a clearing, at the top of a hill, that gives me a better view of where I want to go.  Or a reminder of why I started the blog.

I read an article this week, discussing out-of-school learning vs school learning, and how often children do not connect the two of them (Saxe, 1984). It made me think of one of my first posts  : “For every single worksheet my children may bring from School, I want to make sure they know why they are learning these skills” (see post here). Indeed, whatever we do at home, I always try to connect it to Rosie’s or Tom’s school learning. But it might not be natural for everyone.

As you may have noticed with my lastest posts, I was quite inspired by the conference I attended to in June, on Cognitively Guided Instruction. One of the speakers, Tracy Zagger wrote recently a post for new math teachers (here), wanting them “to become addicted to listening to students’ mathematical ideas”. I am not a new math teacher, but it is definitely how I feel.  I think one of the reasons I am so attracted to the CGI approach is that it deeply echoes my vision of  seeing every child as a unique person and my belief that every child, in a supportive environment, can succeed. After the conference, I started following people on Twitter, exploring new blogs. Some are full of activities to implement in the classroom. Others bring math to the home, with discussions on the spot while cooking dinner, or buying groceries. Whether you browse the web as a parent or as a teacher, you can cross the paths of very inspiring people, and the resources are endless. But I see how a piece of the puzzle can easily be left aside, how the link that connects what is learned/done at school with what is learned/done at home can be forgotten.

I will continue my walk in the woods, I even expect reaching out into some deep dark woods as I begin to embrace my doctorate program tomorrow, but I know for sure that I want to keep focusing my effort on working on that bridge. Connecting both worlds can only take us even further.


Reference

  • Saxe, G. B.  (1988).  Candy selling and math learning.  Educational Researcher, 17(6), 14–21.

 


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.

How Many #2

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 !

 

 

 

 


Solve & Share #1 – Confusing dimes !

Our journey “Solve & Share” will take us to exploring math tasks and sharing some reasoning out loud. I thought the following word problem would be a good way to start as it illustrates, I hope, how much you can learn about your child’s understanding in math by just listening to him/her.

The problem was from the South Dakota Booket I discussed previously (here). As always, my child Rosie, 8, could solve the problem in “any way that makes sense to her” (Carpenter et al, 2014). She could model the problem, with manipulative, a drawing  or a trial error approach, she could use counting strategies, or number facts. As always, it was up to her.

The problem was:

Kenata has 167 coins in her jar. 50 of them are dimes, and the rest are pennies. How many pennies does she have?

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I had seen Rosie solve problems that looked similar to me. Using counting strategies and number facts. And confidence.

With this one, she froze.

At some point, she drew the picture shown on the right, with not much conviction though. Used to the Base Ten Blocks and their representation on paper, she drew the 167 coins as 1 Hundred (i.e. the “gridded square” on the top ), 6 Tens (i.e. the tallies), and 7 Ones (the little squares). She wrote an equation with the unknown number : I was kind of expecting Rosie to finish up.

But she froze again.

She tried with smaller numbers, but it did not seem to help.

I suggested another strategy she had been using successfully in the past when she is stuck : change the story. We talked about pets, dogs and cats, instead of coins, dimes and pennies. Rosie did not have any issue to solve it.

But when she went back to the initial problem, she… froze again.

We went back to her drawing. At this point, however, I noticed through her explanation that the Tens in her jar no longer represented 10 coins but… 10 cents i.e. … 1 dime. No wonder why she was confused. Dimes and pennies are so often associated to cents in word problems, that she could not see them as just coins anymore.

I could have helped her, and said “Rosie! Your Tens represent Tens of coins, not Tens of cents!”.

But I did not.

Because I rarely do. Following the steps of Cognitively Guided Instruction (Carpenter et al, 2014), I prefer waiting that it comes from her, even if it requires an additional 5-10 minutes. Or more. But little step by little step, going back and forth from her drawing to the problem, from the problem to the drawing, she saw it. At some point, she saw where her confusion came from. And provided an answer of 117 pennies almost instantly. With a priceless expression on her face.

It may take time to let a child fully make sense of a problem, or a math concept. But, to me, as a parent as much as an educator, it seems so worth it.


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.

 

 


“1+1=5”, by D. La Rochelle & B. Sexton. It is all about the units !

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Last month, I attended a presentation about units (Cipparone & Bass, 2017). When C. Danielson (“Talking Math with Kids”) mentioned the book “1+1=5”, I quickly wrote the title on a Post’It, knowing that as soon as we were back home,  I would check it out.

I am so glad I did. Such a fun support to make children think about units.

Each page presents a drawing and an equation, such as a unicorn and a goat and “1+1 = 3?”. On the next page, the equation includes the units i.e. 1 unicorn + 1 goat = 3 horns. Indeed, 1 + 1 = 3 :-)

You may have read it in some of my previous posts, I always remind my daughter Rosie, 8, to provide the unit at the end of a word problem, and even invite her to write the units in her equations. This book was just perfect to reinforce my point, and led us to an instructive talk about the importance of the units.

Rosie LOVED that book, and could not stop talking about it for a week, finding new examples on her own. In fact, if you meet a little girl who claims, with a mischiveous grin, that “1+1 = 3”, enjoy: you may have just met Rosie :-)


Reference:

Peter Cipparone & Hyman Bass, 2017. Bringing Out the “Unit” Across Mathematical Domains. Cognitively Guided Instruction. 2017 National Conference, Seattle June 26-28.


“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 !

FullSizeRender

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
  • Addition
  • 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. FullSizeRender2The 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.

 

 


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).Summer Bridge
  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.
    • Practicing additionI 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 Subtractioncome 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 !
    • Word ProblemI 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. MathFairy


* 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.