When I was in school (Rubik’s Cubes and parachute pants were all the rage—you do the math), teachers taught us how to solve problems. (Sounds reasonable, right?) They would introduce a problem, show us step by step how to solve it, and then give us 20 others just like it to do on our own. All was good—until, that is, I ran across another kind of problem. I would try to solve that problem the same way, with the same “steps”—usually with disastrous results. I wanted everything to fit into a nice little algorithm, and if it didn’t, I was lost.

I have long since forgotten all of those steps and most of those algorithms. But I am very good at problem solving. I use my own strategies—I draw a picture, I guess and check, I work backwards. It isn’t always pretty, but it works.

That’s pretty much the model Larry Buschman advocates in his book, Share and Compare. As teachers, we do not teach or model solutions to a problem. We present the problem, and then guide the children through probing questions as they solve the problem themselves. There is no right or wrong way to solve the problem. Students are encouraged to try new strategies. Afterwards, students share their solutions in Mathematician’s Chair (and learn from each other).

A typical Share and Compare math lesson includes:

1. A Warm-Up activity: Students learn and practice math conventions.

2. The Problem-of-the-Day: Students solve the problem individually or in groups.

3. Mathematician’s Chair: Students share their solutions and give feedback to others.

4. Compare: Students compare their solutions, examining similarities and differences.

When I was first presented with this model, I was skeptical that kindergartners would be able to solve problems without modeling, or would be able to effectively communicate their solutions in a manner that would be helpful to others. But a year-and-a-half later, I’m coming around. It takes a lot of patience and persistence—on both the students’ and teacher’s part! But if you stick with it, I promise you will be convinced as well.

Comments

You can follow this conversation by subscribing to the comment feed for this post.

I saw your post on Proteacher and followed the link. I am very interested in learning what this is all about. I want to do a math workshop, guided math and/or something other than the way I have been teaching math. I am a very innovative, creative teacher, but I confess even though I use manipulatives in my math instruction, my instruction is still whole group and too traditional. Help!

I LOVE your math problem of the day questions! My teammate and I really hope that you continue posting these questions! We'll be checking back very frequently! :) They are awesome and I am so excited to try this with my kids and see how they think about math! :)

Fabulous Math Journal Information! Just what I have been looking for. I have book marked your website and plan to visit you each week so PLEASE keep the Math Journal ideas coming!!!! Thank you SO much for sharing your ideas with us!

How long does each part of the share and compare lesson take?
-------------------------------------------

Working on the actual problem usually takes the majority of our math time. A typical day might be a 5-10 minute warm-up, 20 minute problem, 10-15 minute mathematician's chair. But that's not set in stone--somedays we do a longer whole-group warm-up and a shorter problem.

I ordered the Buschman book and have integrated the share and compare work into my weekly math schedule. Unfortuately, I only have time to do it one day a week due to the new math program requirements, but it will be a huge help. Thanks so much for all your generosity and creative, practical and concept-centered math information!

I just discovered you site and love it. Although it looks like there are great resources for the start of the year, have you posted recent problems or activities for winter/spring content?

I have been searching and searching for a math journal and finally I have found something great. I hope you keep posting about the journal...cant wait to use it. Thanks for sharing.

I love your questions. I have been wanting to try a problem solving program in my classroom and was looking for a starting point. I encourage you to continue posting your questions!

Hi All,
I found a curriculum that I am excited about called INVESTIGATIONS. It is similar to the Share and Compare book you presented Katheryn, in that the focus is on problem-solving and articulation of concepts. It is not just "doing math" - it includes the important piece of being able to tell "why" you got the answer you did! I am anxious to get the Buschman book as a support. It sounds like these approaches would mutually support each other!One thing I value in your posts Kathryn is your honest evaluations...knowing the ups-and-downs helps us to proactively create solutions for the downs BEFORE we have 20 bazillion (a new math term :D )smiling, ready to learn children surrounding us! I will keep you posted on its success. Thank you for your simple lesson plans - they perfectly amplify what needs to be done in class. Many, many thanks, Kathryn! Your appreciative new friend!(YANF <3)

This is such a good idea. I teach special needs preK this year (but they are all 5 - need another year in PreK) and I was looking for some way to use notebooks like this. I am going to try to come up with even easier problems to start with. If I can do it I will post them on my website and share them with you. Thanks!! I am also going to put you on my blogroll.

Thank you so much for posting your math Problem Solving ideas! I have so many questions to ask you but I am unable to e-mail you! Each time I try, it will not go through! Could you possibly e-mail me your e-mail address? My e-mail is brooke_mountain@hotmail.com. Thanks again! I poured over your blog last weekend and now am busy re-vamping my math activities! It was just the inspiration that I needed! :)

I love your website!!! I am a teacher in Texas and your ideas are perfect, since we are required to teach the same TEKS. I have begun to use several of your ideas. I love the science notebook ideas and used your lessons for introducing the notebooks. It went very well. I want to implement the math problem solving notebooks next. Thank you for sharing all your wonderful ideas for free. Most sites want a subscription. Please keep the ideas coming. I am so thankful to have found your site. Bless you!!!!

I LOVE all of your ideas!! Thank you so much! Do you have a list of each of your Math questions? Our curriculum is in a different order than you and I would love to use the topic questions that I am teaching now. We do not teach patterns until the second nine weeks. If you have these I would LOVE them! My team is struggling with Math. Thanks so much!!

I also LOVE all of your problem solving ideas...would you be willing to share a list of your Math questions? We currently have Saxon Math which does not challenge our kiddos so finding your sight was a Godsend :). Thank you!

The letters and numbers you entered did not match the image. Please try again.

As a final step before posting your comment, enter the letters and numbers you see in the image below. This prevents automated programs from posting comments.

I saw your post on Proteacher and followed the link. I am very interested in learning what this is all about. I want to do a math workshop, guided math and/or something other than the way I have been teaching math. I am a very innovative, creative teacher, but I confess even though I use manipulatives in my math instruction, my instruction is still whole group and too traditional. Help!

Posted by: Annette Burns | 07/14/2010 at 11:00 AM

I LOVE your math problem of the day questions! My teammate and I really hope that you continue posting these questions! We'll be checking back very frequently! :) They are awesome and I am so excited to try this with my kids and see how they think about math! :)

Posted by: Kristy | 07/16/2010 at 11:25 PM

Fabulous Math Journal Information! Just what I have been looking for. I have book marked your website and plan to visit you each week so PLEASE keep the Math Journal ideas coming!!!! Thank you SO much for sharing your ideas with us!

Posted by: Robin | 07/20/2010 at 08:27 PM

I love this.... But, I am not sure I would be able to come up with my own problem of the day. Do you have a book?

Posted by: Christopher Gibbs | 07/21/2010 at 08:30 AM

Same as the above comments! Please keep sharing ideas! :) My students will love this.

Posted by: Susan | 07/24/2010 at 07:57 PM

How long does each part of the share and compare lesson take? Btw your science badge idea is awesome!

Posted by: Christie Neise | 07/24/2010 at 08:59 PM

Found the book at NCTM for $24.95 and it's on my wish list.

Thanks for the great info!

Posted by: Kris | 07/25/2010 at 06:16 PM

How long does each part of the share and compare lesson take?

-------------------------------------------

Working on the actual problem usually takes the majority of our math time. A typical day might be a 5-10 minute warm-up, 20 minute problem, 10-15 minute mathematician's chair. But that's not set in stone--somedays we do a longer whole-group warm-up and a shorter problem.

Posted by: Kathryn | 07/26/2010 at 09:27 PM

please share the problems you will be using this school year...iam eager to learn and do...

Posted by: CKS | 07/27/2010 at 02:08 AM

What math program do you use? I want to purchase the program or the math activities you use. Thanks

Posted by: Shannon Bowden | 07/27/2010 at 11:03 AM

I ordered the Buschman book and have integrated the share and compare work into my weekly math schedule. Unfortuately, I only have time to do it one day a week due to the new math program requirements, but it will be a huge help. Thanks so much for all your generosity and creative, practical and concept-centered math information!

Posted by: Kris | 08/03/2010 at 01:26 PM

Amazing!!! I love your ideas, please post more :)

Posted by: Kerry | 10/24/2010 at 11:40 AM

I just discovered you site and love it. Although it looks like there are great resources for the start of the year, have you posted recent problems or activities for winter/spring content?

Posted by: Patty | 03/07/2011 at 08:58 AM

I want to do a math workshop, guided math and/or something other than the way I have been teaching math.

Posted by: ClubPenguinCheats | 03/28/2011 at 09:13 PM

I would love to print out the math problems on labels even if it is at a cost . Where can I get the labels that others are talking about?

Posted by: Kathryn sarchuk | 08/02/2011 at 09:43 PM

I have been searching and searching for a math journal and finally I have found something great. I hope you keep posting about the journal...cant wait to use it. Thanks for sharing.

Posted by: marisol | 08/05/2011 at 01:25 PM

I love your questions. I have been wanting to try a problem solving program in my classroom and was looking for a starting point. I encourage you to continue posting your questions!

Posted by: Amanda | 08/06/2011 at 09:47 AM

Hi All,

I found a curriculum that I am excited about called INVESTIGATIONS. It is similar to the Share and Compare book you presented Katheryn, in that the focus is on problem-solving and articulation of concepts. It is not just "doing math" - it includes the important piece of being able to tell "why" you got the answer you did! I am anxious to get the Buschman book as a support. It sounds like these approaches would mutually support each other!One thing I value in your posts Kathryn is your honest evaluations...knowing the ups-and-downs helps us to proactively create solutions for the downs BEFORE we have 20 bazillion (a new math term :D )smiling, ready to learn children surrounding us! I will keep you posted on its success. Thank you for your simple lesson plans - they perfectly amplify what needs to be done in class. Many, many thanks, Kathryn! Your appreciative new friend!(YANF <3)

Posted by: Mrs. B :D | 08/07/2011 at 01:15 PM

I look forward to seeing your questions. I love the idea of using labels for the problems especially at the beginning of the year.

Posted by: Theresa | 08/16/2011 at 10:17 PM

This is such a good idea. I teach special needs preK this year (but they are all 5 - need another year in PreK) and I was looking for some way to use notebooks like this. I am going to try to come up with even easier problems to start with. If I can do it I will post them on my website and share them with you. Thanks!! I am also going to put you on my blogroll.

Posted by: Kristin Whiting | 08/21/2011 at 06:12 PM

Thanks for sharing!You have great ideas.

Posted by: kay | 08/25/2011 at 05:13 PM

Thank you so much for posting your math Problem Solving ideas! I have so many questions to ask you but I am unable to e-mail you! Each time I try, it will not go through! Could you possibly e-mail me your e-mail address? My e-mail is brooke_mountain@hotmail.com. Thanks again! I poured over your blog last weekend and now am busy re-vamping my math activities! It was just the inspiration that I needed! :)

Posted by: Brooke Mountain | 08/31/2011 at 08:23 PM

I love your website!!! I am a teacher in Texas and your ideas are perfect, since we are required to teach the same TEKS. I have begun to use several of your ideas. I love the science notebook ideas and used your lessons for introducing the notebooks. It went very well. I want to implement the math problem solving notebooks next. Thank you for sharing all your wonderful ideas for free. Most sites want a subscription. Please keep the ideas coming. I am so thankful to have found your site. Bless you!!!!

Posted by: Brenda Lewis | 09/12/2011 at 07:59 PM

I LOVE all of your ideas!! Thank you so much! Do you have a list of each of your Math questions? Our curriculum is in a different order than you and I would love to use the topic questions that I am teaching now. We do not teach patterns until the second nine weeks. If you have these I would LOVE them! My team is struggling with Math. Thanks so much!!

Posted by: Emily Puente | 09/17/2011 at 04:53 PM

I also LOVE all of your problem solving ideas...would you be willing to share a list of your Math questions? We currently have Saxon Math which does not challenge our kiddos so finding your sight was a Godsend :). Thank you!

Posted by: Megan Flohr | 11/30/2011 at 01:33 PM