Category Archives: Cognitively Guided Instruction

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.

 


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.

 

 


“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 !

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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.

 

 


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®.HowMany#1

“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.

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.

HowMany#3

  • “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.

MoreSummerBooks

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.

 

 

 

 

 


Exploring word problems throughout Summer

Summer break is here, and we are back to exploring word problems regularly.

Here is a good source of word problems if you want to do the same:

South Dakota Booklet

As always with our math journeys (e.g. Time 4 Fractions or WedWordPro), I simply invite my child Rosie, 8, to solve a problem in a meaningful way to her (Cognitively Guided Instruction, Carpenter et al, 2014), and share her thinking out loud. Drawing a visual representation on paper to make sense of the problem, using manipulatives (e.g. buttons, Legos®, Base Ten block, flashcards to fold and cut, etc), writing an equation and solving the problem using a strategy of her choice, it is up to her, I just listen :-)

Enjoy !


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.

Update Time 4 Fractions – Problem #5 – Peg dolls

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall. I am updating the posts from last year with videos, in case you want to join us this yearClick here if you want to know more about the journey and the previous problems.

Here is Problem #5, a partitive division problem. Last week, with the measurement division problem, children knew the number of items in each group, and needed to find the number of groups. This week, children know how many groups they have, and have to find out how many items are in each group. Just another way to keep exploring division and mathematical relationships.


Time 4 Fractions –  Problem #5 – Peg dolls

PegDollsLevel Yellow – Peter and Julie made 6 peg dolls. They put them into 3 gift
bags with the same number of peg dolls in each bag. How many peg dolls are in each bag?

Level Orange – Peter and Julie made 18 peg dolls. They put them into 6 gift bags with the same number of peg dolls in each bag. How many peg dolls are in each bag?

Level Red – Peter and Julie made ___ peg dolls. They put them into ___ bags with the same number of peg dolls in each bag. How many peg dolls are in each bag?


As always, invite your child to solve one of the problems by

  1. modeling the problem with manipulatives (such as buttons, marbles, etc, and small containers),
  2. representing the problem on a piece of paper, and/or
  3. writing an equation.

When your child is done, invite him/her to share his/her reasoning with you.

With level Red, I left again the option open to pick the number of peg dolls and the number of bags, as my child seems to enjoy the freedom. You may want to invite your child to explore Level Yellow or Level Orange first, though, with modeling the problem with manipulative or a picture. Be aware though, that depending on the numbers the child picks, Peter and Julie may have some peg dolls left (e.g. 13 dolls to put into 5 bags), or may not have enough dolls (e.g. 6 dolls, to put into 12 bags). Let me know how it works !

Sharing my experience (Fall 2015)

My child solved Level Yellow first by modeling it, though dispatching 6 marbles into 3 containers, one marble at a time. She also did a representation of the problem, and wrote an equation (repeated subtraction). Problem#5

For Level Red, she picked 20 peg dolls, and 4 bags. Then, she asked me to solve it. But I am glad she did, as we ended up talking about how different people may use different ways to solve a same problem, and how she will learn additional strategies and symbols at school (i.e. division instead of repeated subtraction, multiplication instead of repeated addition).

Sharing my experience (Fall 2016)

Here is Rosie exploring Level Orange with buttons. As always, it is just to provide an example of how a child may explore the problem.

Enjoy !


Reference:

Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.