# Category Archives: Reasoning

## Our 3rd edition of Time 4 Fractions is ready to start: all aboard !

I am quite excited about starting our 3rd edition of Time 4 Fractions in the coming weeks. I should be able to update the posts from last year significantly, since I went to the conference dedicated to Cognitively Guided Instruction in June and I am taking a course on Children’s thinking this semester as a doctoral student.

As you may remember, I started Time 4 Fractions two years ago, after I read the book “Extending Children’s Mathematics – Fractions and Decimals” (Epson & Levi, 2011) as a M.Ed. student, thinking “This IS the way I would have liked to explore fractions! “. An ah-HA! moment, a true eye-opening: building up meaning for fractions through equal sharing problems. A wonderful approach to pursue at School. But also at home, I believe: the more opportunities to extend math reasoning, the better.

Over the twelve coming weeks, I am going to post a word problem that will take the kids to slowly, gradually, explore the concept of fractions. We will start our journey with multiplication problems (yes, even with lower graders, click here if you are not sure why !), division problems, then, finally equal sharing problems, the core of our journey, and the true beginning of our fraction exploration. The sequence of problems is based on the reading of two books, Children’s Mathematics (Carpenter et al, 2015) and Extending Children’s Mathematics – Fractions and Decimals” (Epson & Levi, 2011).

Whether your child is in lower grade or upper grade, I hope you join us. I share what I do with my own child as a illustration of what a child may do, but by no mean as what a child should do. It is not a test, it is not a race. Week after week, problem after problem, children practice their reasoning skills by creating their own strategies to solve problems.

In the previous year, I found it quite convenient to put together a “math box”. You may want to do the same before we start !

• paper and pencils. Markers are also helpful to connect a visual representation to an equation.
• manipulatives to model the problem.  You do not need the base-Ten blocks. Marbles, buttons can do the trick. I like Legos® and Duplos®, as you can stack them in Tens.
• Containers (e.g. paper cups, Tupperware®), to model problems involving groups of items.
• A stack of paper (e.g. blank flashcards), to explore fractions, by cutting parts of a whole, and putting them back together.

Most important, I will be here to support you in the journey. Please, feel free to comment or email at journey2helpchildrenwithmath(at)gmail(dot)com if you have any question about our journey. The more feedback I receive, the more complete the next post will be ! Let’s build up a community of people supporting at home what our children learn during Math instruction !

Off we go !

References:

• Carpenter, T., Fennema, E., Franke, M., Levi, L. and Empson S. (2015). Children’s Mathematics, Second Edition: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. ISBN-13:978-0325052878.
• Empson, S. E., and Levi, L. (2011). Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH : Heinemann. ISBN-13: 978-0325030531.

## 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 : welcome aboard !

My daughter and I went on a 12 weeks journey in the Fall 2016 to explore Fractions. We are doing it again this Fall. I am updating the posts from last year, in case you want to join us this year.

I have little memory of studying fractions at School. I remember adding fractions, multiplying fractions, finding the least common denominator, but nothing about exploring the concept as such. It was last year, while I was taking a class about fractions, reading “Extending Children’s Mathematics – Fractions and Decimals”(Epson & Levi, 2011)  that I started thinking: “Ah ! This IS the way I would have liked to explore fractions !”. An ah-HA! moment, a true eye-opening. Using word problems to build meaning for fractions. Then, incorporate symbols and equations. A wonderful approach to pursue at School. But also at home, I believe: the more opportunities to extend math reasoning, the better.

How is it going to work :

• Once a week, I will invite my child to explore a word problem and share my experience with you. We will start our journey, labeled as “Time 4 Fractions”, with multiplication problems (yes, even with lower graders, click here if you are not sure why !), division problems, then, finally equal sharing problems, the core of our journey, and the true beginning of our fraction exploration. The sequence of problems is based on the reading of two books, Children’s Mathematics (Carpenter et al, 2014) and Extending Children’s Mathematics – Fractions and Decimals” (Epson & Levi, 2011).
• Each problem is differentiated to target all elementary grades and is quite short. A child may be done within 5-10 min, or may decide to take a few days to fully explore it with a visual representation and manipulatives. It is not a test, it is not a race. Week after week, problem after problem, children strengthen their reasoning skills by creating their own strategies to solve problems.
• I found it quite convenient to put together a “math box”. You may want to do the same before we start !
• paper and pencils. Markers are also helpful to connect a visual representation to an equation.
• manipulatives to model the problem.  You do not need the base-Ten blocks. Marbles, buttons can do the trick. I like Legos® and Duplos®, as you can stack them in Tens.
• Containers (e.g. paper cups, Tupperware®), to model problems involving groups of items.
• A stack of paper (e.g. blank flashcards), to explore fractions, by cutting parts of a whole, and putting them back together.
• Most important, I am here to support you in the journey. Please, feel free to comment or email at journey2helpchildrenwithmath(at)gmail(dot)com if you have any question about our journey. The more feedback I receive, the more complete the next post will be ! Let’s build up a community of people supporting at home what our children learn during Math instruction !

Off we go !

References:

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

## Small, 2012 – Reaching out to all children

Here is another book I studied as a graduate student that I found interesting to share.

Small, M. (2012) Good Questions: Great Ways to Differentiate Mathematics Instruction, 2nd Ed., Teachers College Press, NY.

The author suggests 2 different types of math tasks to reach out children with different skills and needs. Makes sense in a classroom, of course, but it makes sense to me at home as I start seeing my son Tom, 4, willing to “do math” with his sister Rosie, 7 (see my previous post here on Doing Math outside, for instance).

Open questions:

The task is “framed in such a way that a variety of responses or approaches are possible” (Blanton, p6). Remember my post on Vygotsky  (here) ?  Well, the goal is to design the task “in the appropriate zone of proximal development for all students” (Blanton, p6), so that every student can be part of the discussion.

Here is an example of what we did recently:

“Go outside and take a picture of a pattern”.

Rosie came back with a pattern found on  a flower, while Tom came back with a  pattern he created with rocks and pine cones. Still, we were able to discuss patterns all together.

It is a set of tasks that children can choose from, that are close enough to be discussed at the same time.

For instance, this afternoon, I asked Tom and Rosie to create a story out of:

• Choice 1: 10 dinosaurs
• Choice 2: 3 cars

Again, even if Tom used a number smaller than Rosie to create his story, they still were able to share what they did with each other. Also, Rosie was able to create a math problem, while Tom invented “just” a story involving 3 cars.

Of course, with Tom and Rosie’s difference of age/skills/grade, I may not always be able to provide them with tasks they can explore together, but I really like the idea, and will come back to it regularly.

You may want to  check out the book, too ! It includes hundreds of Open Questions and Parallel Tasks organized by math concepts and grade levels.

## WedWoPro #14 – Last one !

Every Wednesday, I give a chance to my child to explore a word problem a way that makes sense to her. And every Wednesday, I share the word problem, and my experience with you, so you can do the same ! Click here  to start from the beginning ! Hope you join us !

Here comes WedWoPro #14, that will end our journey for a little while. I don’t know if Rosie is getting tired with the end of the year but I think it is time to take a little break in word problems to come back even stronger once Summer break starts.

WedWoPro #14 – Last one

Today, you are the teacher ! Write a word problem, and I will solve it !

Sharing my experience:

Here is what I had to solve :

“Once upon a time, 6 butterflies came to have an ice-cream at a friend’s house because it was very hot outside. 4 more came at the friend’s house to have ice cream.

How many butterflies were there in total?

One butterfly left. Then, 3 more left. Then, 3 more left again.

How many butterflies were still at the friend’s house?”

I love doing this kind of task, and opening the door to creativity. It always lead to fun discussion. Today, it gave us a chance to discuss again how to select information that was necessary to solve the problem (e.g. 6 butterflies, 4 more, etc)  vs the information that was not (e.g. it was very hot outside). It was also a fun way to model how I would solve the problem, and share my reasoning, by pretending to be the kid.

Until next time !

## WedWoPro #13 – Few more days !

Every Wednesday, I give a chance to my child to explore a word problem a way that makes sense to her. And every Wednesday, I share the word problem, and my experience with you, so you can do the same ! Click here  to start from the beginning ! Hope you join us !

Here comes WedWoPro #13. We will be working with time for another week. Just to show Rosie that time is not only about minutes and hours. A one-step problem.

WedWoPro #13 – Few more days !

Rosie was very excited. Soon, his grandma would come to visit.

• “I can’t wait to see her tomorrow ! “
• “I can’t wait to see her  ! Just 2 more days !”
• “I can’t wait to see her ! Just 3 more weeks !”

If today is Wednesday, April 13, when is Grandma coming ?

Here is the .pdf if you want to print it out (WedWoPro13).

• Level Green:  Grandma is coming on Thursday, April 14.
• Level Orange: Grandma is coming on Friday, April 15.
• Level Red: Grandma is coming on Wednesday, May 4.

As always, invite your child to solve the level of his/her choice a way that is meaningful to him/her (Carpenter et al., 2014) !

Sharing my experience:

Well, Rosie’s grandma actually arrived TODAY from France. Too much excitement around here… Will have to update the post later !

A bientôt !

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.

## Making sense of subtracting in column

I thought I should complete my latest post,  making sense of adding in column (here), with a quick post on using Base Ten blocks to make sense of subtracting in column. My daughter is not there yet, but your child may be.

The picture above presents a concrete illustration of 43 – 15. It is quite helpful for kids to visualize that, when subtracting 15 to 43, they trade a Ten from the Ten column into Ones. Then, they can subtract 5 Ones to 13 Ones, 1 Ten to 3 Tens, and end up with 2 Tens and 8 Ones i.e. 28.

I just spent the morning in my son’s classroom. He is 4 and attends a Montessori school. Their approach to teaching math is amazing. Indeed, using concrete objects, little 4-5 years old kiddos solve 4-digits addition without even thinking about it.  I should write a special post on the Montessori approach to Math. Quite inspiring, indeed.