Tag Archives: cognitively guided instruction

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.

 

 

 

 

 


Update Ending Time 4 Fractions – Problem #12 – Sharing cereal bars

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

Here comes our last Equal Sharing problem !


Time 4 Fractions –  Problem #12 – Sharing cereal bars

Level Yellow – 2 people want to share 1 cereal bar so that each of them gets the same amount. How many cereal bar would each get?

Level Orange – 3 people want to share 1 cereal bar so that each of them gets the same amount. How many cereal bar would each get?

Level Red – 5 people want to share 3 cereal bars so that each of them gets the same amount. How many cereal bar would each get?


Invite your child to either model the problem (with paper and scissors for instance) and/or represent the problem with a picture. If your child has learned about fractions at school, invite him/her to connect symbols to the model or picture. And as always, invite your child to share his/her reasoning with you !

The problem will lead to a answer of each person getting 1/2 of a cereal bar (level Yellow) , 1/3 of a cereal bar (level Orange) or 3/5 of a cereal bar (level Red).

problem12

Level Red – Child’s sample

This is the last problem. What can you do now ?

The goal of T4F was to provide children with opportunities to explore fractions at home, so they have stronger foundations to build up on when they study fractions at school. This is our last problem, but it does not have to be the end of our journey. The set of problems was designed to provide a wide range of answers, to explore halves, fourths, thirds, fifths and so on, so do not hesitate to go back to these problems and provide one regularly to your child, until your child figures out that “a thing shared by b people is a/b” (Empson & Levi, 2011, p25).  For instance, Problem 12, Level Orange, leading to an answer of 1/3 would be an instructive step towards Problem 8, Level Red, that leads to an answer of 2/3.

The level of difficulties can be seen as follow (Epson & Levi, 2011):

  • Equal Sharing problems that lead to a whole number (i.e. Problem 8, Level Yellow)
  • Equal Sharing problems that lead to an answer that is more than one, with the children having to decide what to do with any left over they may have (first in halves, e.g. Problem 8, Level Orange, or Problem 9, Level Yellow, then fourth e.g. Problem 9, Level Orange)
  • Equal Sharing problems that lead to an answer that is less than one (first with halves or fourths e.g. Problem 12, Level Yellow, then thirds, e.g. Problem 8, Level Red, Problem 12, Level Orange, and so on)

I am including a table summarizing the problems and set of numbers we have exploring so far, I thought it might help.

Problem Level Number involved
Problem 8 – Sharing paper Level Yellow 2
Level Orange 2 1/2
Level Red 2/3
Problem 9 – Sharing bananas Level Yellow 2 1/2
Level Orange 1 1/4
Level Red 4/5
Problem 10 – Sharing apples Level Yellow 1 1/2
Level Orange 2 1/4
Level Red 4/6
Problem 11 – Sharing clay Level Yellow 3 1/2
Level Orange 1/2
Level Red 3/8
Problem 12 – Sharing cereal bars Level Yellow 1/2
Level Orange 1/3
Level Red 3/5

Hope you enjoyed our T4F journey ! As always, I  appreciate any feedback you may have. Comment, or email at journey2helpchildrenwithmath(at)gmail.com.


Reference

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


Update Time 4 Fractions – Problem #10 – Sharing apples

My daughter and I went on a 12 week-journey last year to explore Fractions. We are doing it again this Fall/Winter. I am updating the posts, 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 the problem for the week.


Time 4 Fractions –  Problem #10 – Sharing apples

Level Yellow – 2 people want to share 3 apples so that each of them gets the same amount. How many apples would each get?

Level Orange – 4 people want to share 9 apples so that each of them gets the same amount. How many apples would each get?

Level Red – 6 people want to share 4 apples so that each of them gets the same amount. How many apples would each get?


As always, invite your child to either model the problem (with paper and scissors for instance) and/or represent the problem with a picture. If your child has learned about fractions at school, invite him/her to connect symbols to the model or picture. And as always, invite your child to share his/her reasoning with you !

Level Yellow leads to 1 apple and a half, Level Orange leads to 2 apples and a 1/4 of an apple, and Level Red leads to 4/6 of an apple, or its equivalent 2/3, depending on the strategy the child may use.

Sharing my experience (Fall 2015)

My child went with Level Yellow and Level Orange. I was surprised to see her writing a fraction symbol (1/4). She apparently learned the symbol on her own while playing an education game on the tablet, through a short video, showing a pizza, cut into halves, fourths, and eights. Pretty neat, but at one point, the video talks about 3 fourths of a pizza (3/4) left to eat showing … 6 eights of a pizza (6/8). The 2 fractions are equivalents, but how puzzling to hear 3/4 and see 6/8 of a pizza ? 
 Pb10orange

Sharing my experience (Winter 2017)

There is a significant gap between Level Orange and Level Red, so of course, it is perfectly fine if a child decides to explore only Level Yellow and/or Orange. My child tried Level Red by sharing the apples in halves, providing half of an apple to each child but having some left over. We will be back next week to explore further !

Enjoy !


Reference

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


Update Time 4 Fractions – Problem #9 – Sharing bananas

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.

Another Equal Sharing problem (Empson & Levi, 2011) to help children make connections with fractions.


Time 4 Fractions –  Problem #9 – Sharing bananas

Level Yellow – 2 children want to share 5 bananas so that each of them gets the same amount. How many bananas would each get?

Level Orange – 4 children want to share 5 bananas so that each of them gets the same amount. How many bananas would each get?

Level Red – 5 children want to share 4 bananas so that each of them gets the same amount. How many bananas would each get?


Invite your child to either model the problem (with paper and scissors for instance) and/or represent the problem with a picture. If your child has learned about fractions at school, invite him/her to connect symbols to the model or picture. And as always, invite your child to share his/her reasoning with you !

Level Yellow involves halves, level Orange, fourths and Level Red, fifths. Level Yellow leads to a mixed number (2 1/2) although it may make more sense to some children to give each child 2 bananas, and have a banana left. Level Orange leads to a mixed number (1 1/4), level Red to a proper fraction (4/5).

Sharing my experience (Fall 2015)
My child got a little frustrated with Level Red. If it happens to your child, you may want to start the problem with 5 children sharing 1 banana.  It was a good alternative for us.

Sharing my experience (Fall 2016)

We continued modeling the different levels with flashcards, through folding/cutting paper similarly to what we did last week, each flashcard representing a banana. It is a good way to explore half, fourth, or fifth, depending on the level, with the option to going back to a “whole” banana if need be. We also took the chance to compare a fourth of a “banana “to half of a “banana”, or “two fourth” of a banana to half of a “banana” (“it is the same!”), etc.

Enjoy !


Reference

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


Update Time 4 Fractions – Problem #6 – Stacking blocks

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 #6, a second partitive division problem.


Time 4 Fractions –  Problem #6 – Stacking blocksT4F_Pb#6

Level Yellow – Emmy has 5 wooden blocks. She wants to make 2 towers as tall as possible, using the same number of blocks in each tower. How many blocks should she use in each tower?

Level Orange – Emmy has 13 wooden blocks. She wants to make 4 towers as tall as possible, using the same number of blocks in each tower. How many blocks should she use in each tower?

Level Red – Emmy has 23 wooden blocks. She wants to make ___ towers as tall as possible, using the same number of blocks in each tower. How many blocks should she use in each tower?


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. If your child only writes an equation, encourage him to represent or model the problem as well, and connect the parts of the equation to the model/representation.

This week, all levels involve a remainder (Level Yellow: 2 blocks/tower, 1 block left; Level Orange: 3 blocks / tower, 1 block left) .

Sharing my experience (Fall 2015)RepresentationProblem6

At week 6 of our Time 4 Fractions journey, it seems that my child has her own routine to solve the problem, through at least 2 Levels. She starts with modeling level Yellow, and usually draws a picture to solve level Orange and/or Red. Then, she adds an equation that would match her drawing. This week was no different. She modeled Level Yellow, and drew the blocks, one at a time, in 4 towers, to solve Level Orange.

Sharing my experience (Fall 2016)

Our experience this week was quite similar to last year. Time to move to fraction problems!


Reference:

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


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.


Update Time 4 Fractions – Problem #4 – Making toys

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 #4, a second measurement division problem.


Time 4 Fractions –  Problem #4 – Making ToysMsButternutt

Level Yellow : Ms. Butternut makes wooden toys. She has 5 wheels. She needs 2 wheels to make a motorcycle. How many motorcycles can she make?

Level Orange : Ms. Butternut makes wooden toys. She has 14 wheels. She needs 4 wheels to make a car. How many cars can she make?

Level Red : Ms. Butternut makes wooden toys. She has 31 wheels. She needs ____ wheels to make a heavy truck. How many heavy trucks can she make?


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.

With all Levels, Ms Butternut has a left over of wheels. (Level Yellow: 2 motorcycles can be made, with 1 wheel left, Level Orange: 3 cars can be made, with 2 wheels left).

When your child is done, invite him/her to share his/her reasoning with you. By now, you know the routine, right ?  :-)

Sharing my experience (Fall 2015)

Last week, my child decided to mostly model with manipulative the problems. I think she was not sure how to represent the problem, and confused with the equation she could use. This week, she seemed more confident in her exploration. Problem4

With Level Orange, she started with drawing 14 wheels, and took away groups of 4 one at a time. With such strategy, she quickly saw the equation that could be associated to her reasoning: a repeated subtraction (which is how division can be seen). She used the left over to make a bicycle, but your child may state that Ms Butternut has 2 wheels left.

With Level Red (31wheels, 6 wheels / truck), she decided to draw tallies (by groups of 5) to represent 31 wheels. Then, as previously, she took away groups of 6, to end up with 5 trucks (and a tricycle i.e 3 wheels left). Now, I do not know how she did not get confused with taking groups of 6 out of her tiny groups of 5 tallies, but she did say along the process that  “maybe using tallies was not such a good idea”. I enjoyed watching Rosie discover on her own that some representations may work better in some situations, and less in others. Indeed, it is going to be up to her to select the most useful one depending on the problem.

I am also sharing below the work of a friend’s child, a 5th grader solving Level Red. In parallel with writing the equation, and labeling each part of it, the child also explained the model she could use to solve the problem.

Pb4_LevelRed

 

Sharing my experience (Fall 2016)

Funny how, from one year to the next, a same problem could lead to another exploration. Last year, Rosie drew all the wheels, and took away groups of them (e.g. a group of 4, while solving Level Orange). This year, she added up the groups of wheels needed for one vehicle (e.g. 6 wheels, like last year, to make a truck) until she reaches the total number of wheels available. Such an interesting way to explore the connections between all operations. Also, I think she enjoyed adding equations afterwards, as she could fully connect every part to the picture.

pb4


Reference:

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