A new puzzle to consider

Recently, a friend introduced me to this game/puzzle, and I wanted to be sure to share it with you.  It might be a great thing to use your PTO money or ask a generous parent to pick up for the class.

The only problem I really have with this game is its name: Genius Star.  I hate for that to cause students to believe they aren't capable of solving it, or vice-versa, that if they are able to solve it, it makes them a genius! 

This puzzle explores spatial relationships which for so many of our students can be a challenge.  It also offers students the opportunity to distinguish pieces they should prioritize as well as a trial in perseverance!

Here's how it works:

Roll the die that come with the game:

Lay the little white triangles onto the corresponding triangles on the board:

Now, I usually solve these on my own, and I don't recommend that it becomes a racing game, but students then take the 11 game pieces and place them on the board to cover in the star.

Students can compare their solutions to see what they did the same or different.

Students can also reflect on what made the puzzle challenging or easy. 

The box comes with two black trays to solve the puzzle on, but you could maybe get more than one game so that more students could work to solve. I find it quite fun, and it highlights the spatial skills of math.  The company makes other similar games that will show up when you follow the above link to this game.

Students can solve the puzzle in pairs and discuss their findings, they can investigate the dice and the regions of the board that each one covers, they could create their own puzzle numbers that they think are unsolvable and ask classmates to prove them wrong, they could create fractional questions about the puzzle or its pieces, or you could use two solutions for the same numbers as a same/different discussion. It seems to me like there are a lot of ways that this little puzzle could be used!

Let me know in the comments if you have found or find other ways to use this puzzle in the classroom!

Removing Ability Grouping while increasing Problem-Solving

One of the biggest missteps that occurs in math core instruction is people trying to organize it the same as they do their reading workshop.  

Traditionally in reading workshop, students are grouped by reading level and then called to the teacher to read and discuss a text at their level.  The teacher's small group instruction is really concentrated during this time when meeting with a small group of students away from the rest of their classmates.

In math workshop, we do not recommend this kind of grouping for students often.  In fact, through the work of Peter Liljedahl, more and more teachers are reading and using the research around VRG: Visibly Random Grouping. In his book, Building Thinking Classrooms in Mathematics, he explains how he realized students are more successful this way.

Visibly Random Grouping can be intimidating to some teachers because it means that they have to give up some of their control.  Not like let the room run crazy loss of control, but letting students work with whomever they match with.  Teachers have spent a lot of time organizing and reorganizing their groups for activities.  The thought of just letting it be random might be a lot.  However, once teachers begin to use VRG, they tend to love the results.

To begin with, the students should witness that the grouping has been random. There are apps that group students randomly(even Dojo), but you can always use playing cards, popsicle sticks, birthdays, or any other method that helps them see that you did not intentionally choose who was going to work together. Liljedahl's research shows that groups of 3 work best in most grades to get all students thinking.  In the primary grades, he recommends groups of 2.

Here are some of the benefits of VRG:

• Increased engagement
• Improved collaboration skills
• Reliance of students on each other more than on the teacher
• Improved student ability to work with anyone and recognize the strengths that classmates possess
• Better flow of learning with students not just sharing in their group but also with others in the room: the feeling of being on the same team
• Perception of the students that the teacher believes they can do it.  They don't have to be placed with a student who can "pull them along" or one that they need to support. No matter who they are with, they will be able to access the problem and move forward.

As teachers begin to move towards more use of VRG, they will find themselves removing labels from students.  This subtle shift in thinking will carry over into the students' views of each other, too.  This video by Jo Boaler offers somemre insight into the benefits of heterogenous grouping.

There are many teaching shifts that Liljedahl discusses in his book.  Visibly Random Grouping is one that is a great first shift for all teachers, and it will help when they decide to give a Thinking Classroom a try.

Building Math Culture

 I often talk to teachers about the importance of building a learning culture in their classrooms.  It is so important that students feel safe and valued in their learning ideas.

Recently, I came upon this website, which gives tips and has videos to guide teachers and families in promoting a growth mindset in students.  I especially liked  the examples within the section on Celebrating Mistakes.

It reminded me of a teacher I recently listened to excitedly explain the building of culture in her classroom.  She talked about how she and her students cheer for each other's  mistakes, how students grow in their ability to talk about math through the constant exposure to thinking problems, and how her students love math time in the classroom.  There was no doubt that this excited teacher was talking from the positive experiences happening in her room, and that her students, too, had wonderful experiences as growing mathematicians!

Everyone in the room could feel the joy that this teacher exuded, and I'm sure, like me, they were wishing to have students (or be a student) who were a part of this classroom.  As students grow and content becomes more and more, I think teachers find it more difficult to create this type of atmosphere, but by visiting the website, teachers might find just the motivation to build a stronger classroom learning culture. Even in October it is not too late!

Removing Labels from Students

One thing that often challenges teachers is the way that they group students.  After working with students, they tend to have a group that they consider "low" and a group that they consider "high." Sometimes this designation comes as a result of testing.  This is an area where mindsets need to change.  When we think of students in these terms, we tend to determine their track and limit or expand our expectations--depending on the group.

I understand where this thinking comes from as it was taught to us as best practice for a while.  However, as Jo Boaler and other researchers point out, it is not what is in the best interests of our students as it pigeonholes them into believing that they are not good at math or that math should come easily to them.  Students who are consistently told that math comes easily to them often don't know what to do when it doesn't, and students who are led to believe that math is a struggle for them will want to avoid it because they are "no good" at it.  

So how do we correct this mindset?  There are a number of things that can be done.  First, we need to be very deliberate when referring to students.  We should work to no longer use terms like "high" or "low."  What I find is that most of our students have strengths and weaknesses.  Just because they have difficulty with computation doesn't mean that they don't have a great eye for geometric thinking.  

Secondly,  every effort should be made to not group our students every day by what we perceive to be their ability level.  Students should be heterogeneously grouped so that they can all benefit from hearing the ideas and thinking of others. (Imagine if your principal always grouped teachers according to the strong teachers and the weaker teachers.  How would the weaker teachers ever grow if they only had each other to get ideas from?  And how would the stronger teachers develop better understanding of their craft if they had no one to think deeply with? Everyone brings something to the table.)  Consider using visibly random grouping--an organizational structure researched, practiced, and  encouraged by Peter Liljedahl as helping students become better problem solvers.

Next, use low-floor/high ceiling problems as much as possible with your students.  From your opening routines to the rich tasks you ask your students to explore, find activities that all students can access and take to the level that they want.  These tasks encourage creative thinking and offer may opportunities for rich mathematical discourse.

Being intentional in making these changes will lead to greater learning from our students and improved equity for all.

Looking at Data

While the post title might lead you to believe that this post is about looking at student data, that is not the case in this situation.  Let's look at some resources for students looking at data!

Last fall, I met with some representatives of a major company in my community to discuss math education in our schools.  The number one take-away:  We want workers who know the answer to the question, "What does the data tell you?"  

It is common for teachers to present math pictures or problems to students and ask them "What do you notice? What do you wonder?"  But this post focuses on true data and how we can get students to look at it more thoughtfully, with the ultimate goal being that they can use the information to be better informed about the world around them.

Jo Boaler and youcubed have come up with data talk images!  Here is a link to their page of images (which I am sure will grow...)  Displaying these data images to students not only is great practice with graphs and infographics, but it also allows them to be better readers and understanders of data.  While these might make a great start to a math lesson, they can also be wonderfully embedded into content areas such as social studies and science--all with the simple questions, "What do you notice?"  "What do you wonder?"

Another resource that I often lead teachers to is numberless graphs.  Much like numberless word problems, numberless graphs force students to think and make relationships about the things they know from the graph while still missing key information.  Also much like a numberless word problem, the data is slowly revealed to them to help them make sense of it all in a thoughtful way.

No matter the grade you teach, what are you doing to engage your students in reading and understanding data?  Consider trying some of the links above, and watch the lightbulbs go off!

An exploration for 4th grade

A site that I often frequent for good math inquiry problems is http://www.inquirymaths.org/.



This is a good inquiry I found there for 4th (and 5th) graders.

 24 x 21 = 42 x 12

The link gives good directions and discusses the importance of student conjectures and exploration.  But it leads me to another idea:  How much more valuable is solving this problem and thinking about it deeply than doing a page full of double-digit multiplication?   We sometimes practice the way we were taught, but many times, there are way better ways.  Students, in this case, solve two multiplication problems, look for patterns, develop conjectures, and then work to test those conjectures.  More math problems are being practiced while students are doing some important noticing and thinking.  So much better than a sheet full of double-digit problems.  

Sorts

Sorts are a great way for students to make sense of things around them.  It could be images or numbers or objects or words....the point is to get them thinking about a subject and have them make sense of it in some way.

Here are some ideas for sorts.  Don't think that these are just for Littles.  Older students enjoy them.  They offer a non-threatening way for students to look at things, and they offer us good insight into our students' thinking.  Sorting is very mathematical even when it is not about numbers.  It is about looking at things carefully, finding patterns, and making sense--that's math!

SORT:
pattern blocks
doors
coins
book characters
drain covers
shapes--2D and 3D
numbers (odd/even) (prime/composite) (square/not square) (multiples)
vehicles
words
patterns
emojis
food
volume/area/perimeter
shoes
states
types of graphs (no numbers or titles are necessary)
expressions
angles

Cutting the holiday pie

A good opportunity for exploration and problem-solving for your students.

With one straight cut, you can cut a pie into two pieces. Cutting it again and having the second cut cross the first cut gives you four pieces. With three cuts, you might get up to seven pieces.  What is the maximum number of pieces you can get with 6 straight cuts?

Your cuts do not need to produce equal pieces.  Once students identify the solution, can they keep going with more cuts?  Is there a pattern that begins to emerge?

Photo by Alex Loup on Unsplash

What Does the Data Tell Us?

In a recent conversation with some representatives from a major nationwide business company, we were discussing the needs in elementary math education to help lead students to be productive workers in tomorrow's business world.

The big question they said that students need to learn to answer is, "What does the data tell us?"  We also discussed the importance of probability and statistics,  but in the end, they said that they are looking for workers who can answer this question.

What would this look like in an elementary classroom?   I think it just reframes our questioning.  In many of our activities, we already expose students to real-world information.  We just need to be sure to be more intentional in our questioning in order to get them to look at data.

For example, look at this picture.  I took it thinking that it would be a good one for students to determine what was the best deal. What does it tell us?  What are some possible reasons a person would be willing to pay more for 3 Peep trees when they can get 9 for such a better deal?  

Noticing and wondering is a classroom routine that really benefits our students.  When they notice and wonder, you can ask them what the picture tells them.  Make them infer from the data that they have.  Numberless graphs are a good method of providing data that the students have to make sense of.  Here is a good example of a 2nd grade lesson regarding them.  You can find lots of examples and ideas for using them by reading some of the posts listed here.

What do you notice about this data?  What do you wonder?

We want our students to be able to compute and do basic math, but not at the expense of good math thinking and discussion.  These are the skills that will carry them into the future--not only in their career, but also in their roles as consumers and citizens.

How can you incorporate statistics and probability into your classroom (whether it is a K classroom or a 5th classroom) by framing your questions and your students' thinking around the question, "What does the data tell us?"

3 Act Tasks: Have you tried them yet?

3 Act Tasks offer our students such an engaging opportunity to make sense of math! However, as teachers, sometimes we are afraid to try something new not knowing where it might take us and scared that something bad might happen.  I just heard an analogy recently about this being like being at the top of a rollercoaster:









via ytCropper
And, while it might have twists and turns and dropoffs, not to mention bugs in the face, it also brings an excitement and thrill that we rarely find doing a traditional math lesson. We need to think about putting ourselves in that precarious position at the top of the rollercoaster; it's what we ask our students to do on a regular basis so that they can grow.  We should try it, too!

3 Act Tasks are real-world problem-solving scenarios which require students to make sense of what to do.  During Act 1, they use the reading skills of visualizing, predicting, and inferring in a math context.  They figure out what they need to know to solve the problem. In Act 2, students work to solve the math question in a way that makes sense to them. They discuss their thinking with a friend and compare their answer to their estimate.  Act 3 is exciting because that is when they find out if they were on the right track!  

While they are working during Act 2, you are monitoring (and asking questions that help them understand--not helping too much).  You are monitoring to see the methods that students used in order to solve the problem.  During Act 3, it is your job to have students share ways that they solved the problem.  These should be sequenced so that you can show connections between different methods. Be clear on what your math target for the lesson was and be sure that your models and discussion help that math target to be evident to everyone.  You synthesize the learning at the end of the lesson.

3 Acts are very visual and often use videos to help students better understand the situation. They follow more of the format of "you do, we do, I do" rather than the traditional layout of "I do, we do, you do."

Typically, a 3 Act Task in K-2  about 20-30 minutes.  In 3-5, a task usually takes about 4o minutes.  As students get stronger and more comfortable, the time needed for a 3 Act might decrease.

I am happy to come and model a 3 Act Task for you and your class, but I know that many of you can do them without my support.  You just need to put yourself on the rollercoaster!

At the top of this blog, you will find a tab labeled 3 Act Tasks.  This will take you to a large number of standards-aligned tasks that I have organized using SMART Notebook (and sometimes Google slides).  They are by a variety of mathematicians; I just put all of the pieces together into one format.  You can also find more great tasks by clicking here or here or even by Googling 3 Act Tasks.

As always, share with me your questions, struggles, or successes and let me know how I can help!

Would You Rather? -- Holiday Edition

If you are looking for some problem solving to get your students thinking and proving themselves while mixing in some holiday cheer, maybe these Would You Rather problems will work for you!

Would You Rathers are something that students are very familiar with, and these just include a mathematical twist.  Students can choose whichever option they want, but they need to have a mathematical explanation of why that is what they chose or didn't choose.  They can be used with students of all ages and offer great opportunities for collaboration.

If you like these, you can find more like them at this site, but I am betting you can come up with some good ones of your own!

Holiday Would You Rather 1

Holiday Would You Rather 2

Holiday Would You Rather 3

Holiday Would You Rather 4

Holiday Would You Rather 5

Does it look right? Does it sound right? Does it make sense?

These words were always such a big part of my instruction of reading workshop.  Students used hand signals to help them remember the three questions as they reviewed what they had just read.  Many times, they realized the word they had just read did not sound right or make sense in the context they had just read.  So---they worked to figure out what the new word was that they were struggling to read.

Do we teach our students to do this same kind of thinking when solving a math problem?  In my experience, it is a less common practice.  Sometimes, a teacher will ask the students if the answer makes sense, but really, shouldn't we be teaching our students to ask themselves that question?

Tying our math instruction to real world situations allows for the students to better make sense of their answer.  We need to be intentional in our plans to teach students to make sense of situations and solutions. One of our biggest goals during math instruction is for our students to EXPECT that their answer will make sense.

Continuing to Find Ways to Build Math Talk

I have participated in a number of professional discussions in the last week around the concept of Number Talks.  You may remember that I blogged about Sherry Parish's book before. It is an excellent resource to help you get started with Number Talks.

Many classrooms use Number Talks every day.  It is a great tool for building mathematical discourse, exposure to new strategies, and strengthening students' flexibility and fluency.

However--some people are a little intimidated by Number Talks.  It has some elements of the unknown, and this makes teachers a little leary to try it with their own students.  This is totally understandable.  One way to make this work is for the teacher to work on the anticipation part of the number talk fully.  By anticipating all of the possible ways a student may respond, it will give you the preparation to feel confident as you step into the Number Talk.  It will also allow you time to prepare visual images to represent ways that the students might explain.  It is important that we show the students' ideas using visual representations as this will allow us to reach more students.

I 

Here is a great example of ways that students might see 18 x 5.

I do believe that the term Number Talk is broader than just the use of functions with numbers as is found in the Number Talks book.  Of course, traditional number talks often involve quick-look cards and other visuals to help students visualize the math, but they are still considered to be from the basic concepts presented by Sherry Parrish in her book.

If you click on the tab labeled Routines located at the top of this page, you will find links to resources and videos of a large variety of mathematical routines that will encourage students' math talk.  There are many low-floor/high-ceiling activities which are good to put the students (and sometimes the teacher) at ease when doing a number talk.  In these types of tasks, nearly everyone can find an answer right away, but because these routines lend themselves to multiple answers, we find students recognizing all kinds of things besides the obvious.  Using these type of routines in addition to Number Talks will help your students to grow in ways you won't believe!

Challenge yourself to something new.  Try to add routines to your math class each day--you will be glad you did!

Rethinking Homework

As we rethink the role that homework is playing in our students' learning, we should look at new ways to have our students practice their learning rather than just a set of practice problems each night.  

One idea to consider is a math reflection question for the students to respond to.   Building their metacognition through these written responses has proven to build better mathematical thinkers.  

You might decide to have a menu of questions for students to answer, or you might begin by having them all answer the same question--whatever works best for your students.

Click below to a link of some potential questions.  Reflecting on these ideas each day will help to build the types of thinkers we hope to cultivate.

A shift in math instruction

I'm sure we have all heard a lot about the "shifts in mathematical instruction," but sometimes I think that we get caught up in the shifts in the content of our instruction at the elementary level.  Who teaches time? Where does money get taught?  Third grade doesn't teach multi-digit multiplication anymore?  

The content that we teach has changed some, but in my experience, it is not the hardest shift in our instruction.  The hardest shift in our instruction is explained in the TED talk included below, The Five Principles of Extraordinary Math Teaching by Dan Finkel.  The video is about 15 minutes long, but in it, he highlights some of the importance in this shift for teachers and parents.

These are the principles he highlights:

1.  Start with a question.

2.  Students need time to struggle.

3.  You are not the answer key.

4.  Say yes to your students' ideas.

5.  Play!

Where do you see yourself in these practices?  Can you choose one that you think you can improve?  What supports can you find to help you move forward?

Same or Different?

Looking for a new routine to use in your daily workshop?  Same or Different could be your answer!

This blog has a great variety of images and videos for teachers to use to get this routine started in their classrooms.  There is also a link to a Teaching Channel kindergarten lesson using this routine.

It's just one more way for us to encourage student talk around math.

Estimate! Estimate! Estimate!

A key to building our students' number sense is to practice estimating.  Many times, students and parents look past the estimation part of a computation problem, valuing instead the final, more accurate answer.  However, John Van de Walle's research revealed that estimation is a higher-level skill.  It is important for our students to be able to mentally manipulate numbers through estimation.  It is a real-world skill that they will use daily as an adult.

We know that predictions are important in reading.  It helps readers to focus their comprehension to determine if their predictions are correct.  It also gives them a little more ownership of the story as they read to find connections between the predictions they made and the reality of the story.  As readers grow more proficient, their predictions tend to gain accuracy.  The same is true with mathematical estimates!  Our students will grow in their abilities to estimate the more that they practice.

Math is Fun has a good post that highlights for our students the value of estimation.  It also has a page of estimation games to help students practice in a fun way.

However, for whole class or small group practice, here a couple resources to help you practice regularly in your classroom. 

For K-2:  This Powerpoint has estimation activities for your students to practice.  There are 40 slides in the presentation, and maybe it will give you more ideas to implement this skill into your classroom.

Estimation 180 is a website that I have shared before.  It has images that build off of each other to help students use prior knowledge to make a best estimate.

All of these resources are good for building our students' understanding of number.  What resources have you found helpful in your classroom?  Share in the comments below.

Building Thinkers through Open-Ended Tasks

Open-ended tasks offer great opportunities for our students to grow as problem solvers. They also offer students a non-threatening way to show their work, and develop good vocabulary and justification skills.  

These open-ended problems I created are perfect for December or early January because they are holiday-based.  There are 15 different problems in this document, and I think there is something for every grade level.  Please feel free to download and use whatever pages best fit the needs of your students.  

OPEN-ENDED PROBLEMS

More Real World Math Errors

Kids love to find mistakes that others have made.  The attached file is full of errors found in the real world.  Can your students figure out what is wrong?  

Some of these are geared for older students, but there are some in here for the littles, too!

You can use these as a number talk, an opening activity, a collaborative problem solving activity...just be sure to get the best thinking, can they figure out how to correct it?

If your students like these, here are some from an earlier post.

Click for new file

Websites to help you kick your teaching up a notch!

I have recently come upon these math websites...

I think they can be used in a variety of ways in our classrooms. There are prepared resources on these sites, but they might also inspire you to create your own.  It might also be a good opportunity for students to create them.

Would You Rather?  This site gives two mathematical questions and students have to determine which they believe will give them the better result.

Estimation 180  This site offers opportunities for students to estimate with the hope that practicing estimation more often will build their number sense.

Fraction Talks Images from this site support the teaching of fractions.  Ideas for using these images are limitless, but as a number talk might be a great way to start.

Visual Patterns  The images on this site are the beginning of a pattern.  Can your students figure out how the pattern works?  Can they write an equation to match it?


All of these sites offer opportunities to expand their thinking and especially their thinking about math.  Many of them are low floor/high ceiling tasks meaning that all of your students can be successful with them.  Consider using them with all students!

I would be happy to help you find ways that these activities could work for you and your students.  Just let me know how I can help!