The math homework dilemma – to give or not to give (IF you have the option in your district)? How much to give? To go over it all or only review some of it? What will be most helpful to your students?

Maybe your experiences have been similar to mine: I’ve adjusted my practices from year to year, sometimes spending a lot of time reviewing homework, but other times spending little; some years giving homework related to the lesson, other years giving homework that was basic skills practice; some years lots of problems, other years just a few. There seemed to be pros and cons to each.

Thinking about this topic yet again, I decided to look for some research to see how we can help students get the most out of the homework we assign.

Maybe your experiences have been similar to mine: I’ve adjusted my practices from year to year, sometimes spending a lot of time reviewing homework, but other times spending little; some years giving homework related to the lesson, other years giving homework that was basic skills practice; some years lots of problems, other years just a few. There seemed to be pros and cons to each.

Thinking about this topic yet again, I decided to look for some research to see how we can help students get the most out of the homework we assign.

Research is varied, and opinions about homework are varied; there are books and articles supporting homework, and there are books and articles opposing. For example, research cited in the NCTM article *Making Homework Matter to Students (NCTM, 2017)*, states that there IS a positive correlation between high-quality homework and mathematics achievement (Trautwein 2007 and Dettmers et al. 2010), and that students who completed their homework scored better on assessments. But the studies also showed no relationship between time spent on homework and student achievement. On the other hand, The Program for International Student Assessment (PISA) in 2015 announced that homework perpetuates inequities in education and questioned whether it has any academic value. In __Mathematical Mindsets__, Jo Boaler states that reviewing homework at the beginning of class magnifies those inequities among students. Various other studies have found that homework has a negative effect or no effect on achievement at all.

So, research doesn't necessarily agree on the benefits of having students complete math homework – that’s not all that helpful:-) Especially if your district expects (or requires) you to assign homework.

Research does seem to agree, however, that certain*types* of math homework can provide more benefit to students than others. This is what we're looking for! As stated by Jo Boaler, in __Mathematical Mindsets__, “Research shows that the only time homework is effective is when students are given a worthwhile learning experience, not worksheets of practice problems, and when homework is seen not as a norm but as an occasional opportunity to offer a meaningful task.”

She recommends giving questions that students need to answer in a performance orientation or assigning reflection questions that encourage students to reflect on the math in the day’s lesson and focus on the big ideas, like how the ideas from the lesson could be used in life.

So, research doesn't necessarily agree on the benefits of having students complete math homework – that’s not all that helpful:-) Especially if your district expects (or requires) you to assign homework.

Research does seem to agree, however, that certain

She recommends giving questions that students need to answer in a performance orientation or assigning reflection questions that encourage students to reflect on the math in the day’s lesson and focus on the big ideas, like how the ideas from the lesson could be used in life.

So, let's think about what would help our students get the most out of the math homework we assign.

**1) Assign homework that has a very specific purpose**

That sounds logical, but have you ever been in a hurry and assigned #1-20 on page 47, without really looking at**all** the problems first? I will admit that I’ve been guilty of that. With an assignment like that, students may sense that the purpose was simply to assign homework.

If we’re working on decimal subtraction, it might be better for me to assign 4 problems that require students to remember to annex a zero in the minued or to regroup when there are zeros (since those are the types of problem they often have trouble with) and assign 1 problem that doesn’t require those things. Or, if I want students to think a little more deeply, I might assign 5 error analysis problems and ask them to explain the mistakes in writing. According to the NCTM article, homework assignments like error analysis require deeper thinking and understanding, which is what will benefit our students the most.

**2) Make homework accessible by differentiating**

If students are unable to complete the math homework because it's too difficult (or they believe it's too difficult), there isn't much chance that they'll get any good math practice out of it. And this goes back to the inequities mentioned earlier - if they couldn't do it, what happens to them during the review of the homework? They are likely lost and/or tuning out.

The same applies for the students who find the work too easy - if it's simple for them, they aren't getting good practice or deep thinking. And homework review? They probably find it boring. There are so many ways to differentiate - a great topic for another post:-)

**3) Make homework aesthetically pleasing**

According to ASCD, 2010, if the homework looks uncluttered and is graphically appealing, students may be more interested in completing it. I honestly hadn't thought about this much in the past! But think about your response when you look at a page completely filled with text or too many graphics - how does it make you feel?

**4) Give students the opportunity to discuss their answers**

I have found**great** benefit to giving students time to discuss in small groups. I do this as frequently as possible. It gives me time to circulate and listen to their conversations and questions. And often, students are willing to ask a group member about something that gave them trouble, rather than asking in front of the class. This provides them the opportunity to verbalize their confusion and allow peers to verbalize their understanding of the concepts. This discussion doesn't have to take a lot of time - especially if the homework assignment was only a few problems:-)

**5) Assign h****omework that's efficient**

According to*Five Hallmarks of Good Homework*, ASCD, 2010, this means it probably shouldn’t include cutting things out, gluing them, or creating posters, for example. While I like using "foldables," I'd agree that assigning them for homework may not be the best choice. Where is the math practice in this type of homework?

Based on this information, my action item is to work on creating differentiated math homework assignments that focus on a specific purpose, require deeper thinking, and are graphically appealing.

Do you have any tips to help students get the most of out their math homework?

]]>That sounds logical, but have you ever been in a hurry and assigned #1-20 on page 47, without really looking at

If we’re working on decimal subtraction, it might be better for me to assign 4 problems that require students to remember to annex a zero in the minued or to regroup when there are zeros (since those are the types of problem they often have trouble with) and assign 1 problem that doesn’t require those things. Or, if I want students to think a little more deeply, I might assign 5 error analysis problems and ask them to explain the mistakes in writing. According to the NCTM article, homework assignments like error analysis require deeper thinking and understanding, which is what will benefit our students the most.

If students are unable to complete the math homework because it's too difficult (or they believe it's too difficult), there isn't much chance that they'll get any good math practice out of it. And this goes back to the inequities mentioned earlier - if they couldn't do it, what happens to them during the review of the homework? They are likely lost and/or tuning out.

The same applies for the students who find the work too easy - if it's simple for them, they aren't getting good practice or deep thinking. And homework review? They probably find it boring. There are so many ways to differentiate - a great topic for another post:-)

According to ASCD, 2010, if the homework looks uncluttered and is graphically appealing, students may be more interested in completing it. I honestly hadn't thought about this much in the past! But think about your response when you look at a page completely filled with text or too many graphics - how does it make you feel?

I have found

According to

Based on this information, my action item is to work on creating differentiated math homework assignments that focus on a specific purpose, require deeper thinking, and are graphically appealing.

Do you have any tips to help students get the most of out their math homework?

How often have you gone to a conference and been super-impressed by what a speaker shared? Has it happened often? It happened to me when I went to a conference as a very new teacher (in my second year, I believe), more than 20 years ago. At that conference, I was lucky enough hear Dr. Lola May speak. She was a great presenter, and certainly made an impression on me. I still have the book that was given at that conference and have referred to it many times over the years. It was at this conference that I first learned how to use "casting out nines" to check the answers to multiplication and division problems. I had never heard of this method when I was a student, but being a new teacher, I kind of assumed it was a method well-known to other teachers..... |

until I talked about it during a meeting at which our Curriculum and Instruction director was present. He overheard me explaining it to another teacher; he had never heard of it, was quite surprised and interested in how it worked, and asked me to show him a few more examples.

Over the years, I have taught the method to many classes, and I don't think any students have ever told me that they had already learned it. So, I suppose it isn't as well-known as I had thought (at least not around here...)

The kids really like it because it's a "trick" to check their work (I never taught them*why* it worked - I think that might have been too much for this age). I think it's especially handy for multiplication. Here are the steps of *casting out nines* to check multiplication (you can follow the example on the wheel):

Over the years, I have taught the method to many classes, and I don't think any students have ever told me that they had already learned it. So, I suppose it isn't as well-known as I had thought (at least not around here...)

The kids really like it because it's a "trick" to check their work (I never taught them

1. Going across the rows of the multiplication problem, "cast out" (just cross them out) any 9s or combinations of numbers that add up to 9.

2. Add the remaining digits across each row, until the result is a single digit.

3. Multiply the single digits, and if the result is a 2-digit number, add the digits to get a single digit.

4. Follow the same steps in the product, until you arrive at a single-digit number.

5. If the results match, the answer to the problem is most likely correct (not 100% certain, but most likely); if the results do not match, the product is not correct.

Casting out nines can also be used with the other operations as well, but using it to check multiplication is my favorite.

Have you used casting out nines?

]]>2. Add the remaining digits across each row, until the result is a single digit.

3. Multiply the single digits, and if the result is a 2-digit number, add the digits to get a single digit.

4. Follow the same steps in the product, until you arrive at a single-digit number.

5. If the results match, the answer to the problem is most likely correct (not 100% certain, but most likely); if the results do not match, the product is not correct.

Casting out nines can also be used with the other operations as well, but using it to check multiplication is my favorite.

Have you used casting out nines?

Middle school students still like those fun seasonal activities! Many years ago (I have no idea how many) I used this pattern coloring activity with my middle-schoolers. I don't remember where the idea came from, and I had even forgotten that I ever used it! However, I was looking through an old "November" file to find some ideas for a fun activity for a sub day, and found the tracers and examples in my file.

Once I found it, I DID remember that the kids used to really enjoy this activity. They had fun creating the patterns and deciding what colors to include. So, I gathered materials (graph paper, tracers, colored pencils, thin black markers, construction paper) and left them for the sub, with these directions:

Once I found it, I DID remember that the kids used to really enjoy this activity. They had fun creating the patterns and deciding what colors to include. So, I gathered materials (graph paper, tracers, colored pencils, thin black markers, construction paper) and left them for the sub, with these directions:

1. Students take one piece of graph paper (I use the tiny squares, but younger students could use larger ones).

2. Place the tracer under the graph paper and trace the outline and details of the shape.

3. Color the squares with different shades, alternating light and dark (colored pencils work better than markers).

4. When finished, outline in black, and go over detail lines in a darker color.

5. Cut out and glue onto construction paper.

2. Place the tracer under the graph paper and trace the outline and details of the shape.

3. Color the squares with different shades, alternating light and dark (colored pencils work better than markers).

4. When finished, outline in black, and go over detail lines in a darker color.

5. Cut out and glue onto construction paper.

This is a great way to create attractive pictures any time. I was just so excited to find it again that I thought I'd share:) You can click on the pumpkin or turkey pictures below to download the tracers.

For a more academic activity, I also included the cross number puzzle (pictured above) and a math color by number and Footloose math game. These are both mixed practice and great to use for review any time. You can find these in my TPT store if you click the main image above. Happy coloring!

]]>Having the perfectly-run math class....that's been my goal, year after year. Somehow, in middle school, it has consistently tried to evade me!

In other posts, I've shared that I taught elementary math for years, and always had an hour for math class. That hour gave me the time I wanted to have good warm-ups every day (sometimes taking up half the class with one particular problem that led to additional discussion/extension!); the hour gave me the time to go over homework the way I wanted to. And it still gave me time for a new lesson and practice.

But when I got started teaching math at the middle school, with "44"-minute periods, that was all over. (They aren't really 44 minutes - the students get no time between classes for switching, so switching time comes out of the 44.)

In other posts, I've shared that I taught elementary math for years, and always had an hour for math class. That hour gave me the time I wanted to have good warm-ups every day (sometimes taking up half the class with one particular problem that led to additional discussion/extension!); the hour gave me the time to go over homework the way I wanted to. And it still gave me time for a new lesson and practice.

But when I got started teaching math at the middle school, with "44"-minute periods, that was all over. (They aren't really 44 minutes - the students get no time between classes for switching, so switching time comes out of the 44.)

I tried to use the same kind of warm-up I used in elementary school (a word problem to practice a particular problem solving strategy, including a written explanation). Sometimes these took 20-30 minutes. So, that left only 10 -20 minutes to review homework, teach a new lesson, and practice.....but that didn't work well. So I cut these warm-ups down to once a week and let them take the whole class period. But I felt like warm-ups once a week wasn't enough.

Then I bought a warm-up book (because I really wanted warm-ups each day - it's the best way for me to start my classes). These were shorter (though not always as challenging as I wanted), but so short that some students who got to class first finished before others even arrived (and some of the problems were just too simple). Others just took longer to get done.....so those who were done needed something to do while they waited for the others to finish. Eventually I wrote all of my own warm-ups, so I was very happy with**what** we were covering, but still not happy with the **how**. (One step in the right direction!)

My next issue was reviewing homework. I wanted to go over all (or most) of the problems. I wanted to be sure that I answered all the questions anyone had (and discussed certain problems even if no one asked the questions). So homework often took a long time to go over.

I struggled with the best balance of warm-ups, homework review, lesson, and practice for a couple of years, I have to admit. And no one that I taught with seemed to have the same issues as me. Part of that was because they weren't using warm-ups like I was, so they weren't losing that chunk of time at the beginning of class. But I knew the warm-ups and our discussions were beneficial to the students in the long term.

Here's what I've finally landed on that allows us to use our math classes as efficiently and effectively as possible:

**1) Warm-ups are homework.** My warm-ups are only 2-3 questions per day, so is isn't a long assignment. Even when it's added on to other homework, it doesn't take that much extra time. There are times when students don't have the knowledge to answer a warm-up question (because we may not have learned the concept yet), but they have to at least give it an attempt.

**2) Warm-ups are discussed in groups for the first 5-7 minutes of class. **Students get to class and immediately take out the warm-ups and review the answers with their group members (my students sit in groups of 4-6). This allows for math discussion (love it!); students help each other if someone didn't understand a certain problem. I circulate during this time to listen in, check answers, and help any groups that need help.

**3) When the warm-up discussion is done, students self-check homework** (another 5-7 minutes, depending on # of homework problems). I put all the answers on the board before they come to class, so that as soon as they finish the warm-up discussion, students can start checking their homework. This again gives me time to circulate, check for homework completion and help students that have questions. I normally pick out one or two of the more challenging problems to discuss as a class.

**4) Students prepare for the day's lesson**. For those who get done with the warm-up and homework checking before others, I'll have a question on the board or an activity to begin that pertains to the new lesson for the day. I make it something that isn't necessary to the lesson so that those who took longer with the warm-up and homework won't miss something necessary to the lesson.

**5) New lesson and practice**. Now that warm-ups and homework are down to about 10-15 minutes per period, we have 25-30 minutes for the new lesson and the practice:-)

Do you have 40(ish)-minute math periods? What does your class structure look like?

]]>Then I bought a warm-up book (because I really wanted warm-ups each day - it's the best way for me to start my classes). These were shorter (though not always as challenging as I wanted), but so short that some students who got to class first finished before others even arrived (and some of the problems were just too simple). Others just took longer to get done.....so those who were done needed something to do while they waited for the others to finish. Eventually I wrote all of my own warm-ups, so I was very happy with

My next issue was reviewing homework. I wanted to go over all (or most) of the problems. I wanted to be sure that I answered all the questions anyone had (and discussed certain problems even if no one asked the questions). So homework often took a long time to go over.

I struggled with the best balance of warm-ups, homework review, lesson, and practice for a couple of years, I have to admit. And no one that I taught with seemed to have the same issues as me. Part of that was because they weren't using warm-ups like I was, so they weren't losing that chunk of time at the beginning of class. But I knew the warm-ups and our discussions were beneficial to the students in the long term.

Here's what I've finally landed on that allows us to use our math classes as efficiently and effectively as possible:

Do you have 40(ish)-minute math periods? What does your class structure look like?

I taught elementary school for 12 years and I loved my math centers! They were great. Math class was always an hour, and we had five computers in the classroom, so having a computer center was always an option.

Then I moved to middle school. Math was 44 minutes (minus time for switching classes.....so more like 40 minutes). How could I fit more than two rotations in a 40-minute period?? I longed for block scheduling (our district has never had it)...that would make it so much easier to complete center rotations! For the first year or two of middle school, I kind of gave up on the idea of centers...the activities I wanted students to complete took longer than 20 minutes. So, that would be enough time to finish 2 rotations, IF students started the second they walked in the door and then had no time to clean up/organize at the end of class. But eventually I needed to get my centers back, so I experimented with a few different set-ups before I landed on a structure that works.

Then I moved to middle school. Math was 44 minutes (minus time for switching classes.....so more like 40 minutes). How could I fit more than two rotations in a 40-minute period?? I longed for block scheduling (our district has never had it)...that would make it so much easier to complete center rotations! For the first year or two of middle school, I kind of gave up on the idea of centers...the activities I wanted students to complete took longer than 20 minutes. So, that would be enough time to finish 2 rotations, IF students started the second they walked in the door and then had no time to clean up/organize at the end of class. But eventually I needed to get my centers back, so I experimented with a few different set-ups before I landed on a structure that works.

I willingly admit that I have not found a perfect solution....40 minutes is just too short a time-period for math! However, I've figured out what works for me, and maybe it can work for you, if you also have short math classes.

**My centers are:**

* One per day

* 30 minutes a day

* For 3-4 days, depending on the topic.

**There are many ways to group and assign tasks, but these are the grouping/activity options I normally stick to:**

1) When we only do three days, I create six groups and prepare two sets of materials for some tasks. Students will all complete three tasks over the course of the three days, but they might not all complete the same three...it depends on the topic, their needs, and my goals. The image to the right has two different examples of how I might assign the tasks.

2) When we do four days, I create four-six groups (if my class size is the usual 27-30). If I have only four groups, I usually assign them each a different task and then rotate through those tasks over the four days. If I have five or six, then I'll have two groups complete the same task on the same day, similar to the three-day example.

* One per day

* 30 minutes a day

* For 3-4 days, depending on the topic.

1) When we only do three days, I create six groups and prepare two sets of materials for some tasks. Students will all complete three tasks over the course of the three days, but they might not all complete the same three...it depends on the topic, their needs, and my goals. The image to the right has two different examples of how I might assign the tasks.

2) When we do four days, I create four-six groups (if my class size is the usual 27-30). If I have only four groups, I usually assign them each a different task and then rotate through those tasks over the four days. If I have five or six, then I'll have two groups complete the same task on the same day, similar to the three-day example.

My students are not ability-grouped, so they finish activities at various times. I decided that I have to be happy with some students/groups finishing early and others not finishing (or finishing during a free, non-math period). Doing one center a day, for about 30 minutes allows for some flexibility here. If group members finish early, they can do the following:

1) Finish another center activity, if they had something unfinished.

2) Complete a color by number from our "finished early" resource bin.

3) Use the trivia cards - a set that isn't one of the center activities. (click here to grab this free resource!)

4) Use the pentominoes that we use on the first day of class - there are**always** students who want to complete this challenge.

5) Use a technology source for additional math practice, if we have extras available.

1) Finish another center activity, if they had something unfinished.

2) Complete a color by number from our "finished early" resource bin.

3) Use the trivia cards - a set that isn't one of the center activities. (click here to grab this free resource!)

4) Use the pentominoes that we use on the first day of class - there are

5) Use a technology source for additional math practice, if we have extras available.

Before we start using center rotations, I make sure students have a complete understanding of expected behavior AND of the activities they'll be completing. We complete the different activities together with different concepts at the beginning of the year, and then I use those activities in the centers, using some of the same ones each time.

Students are up and around the room for this, so I typically assign only one or two groups per day.

(with paper and pencil):

This version is a group game, so I only assign one group per day, for less "noise" in the room. They have to talk, and they definitely have fun with this one!

I created a digital version. This allows two different groups to complete the same activity, but one group can use whatever technology we have available. This version could be played by group members independently or in teams.

This is a quiet activity that provides some self-checking practice. Students can also check answers with their group members if they'd like. (They often don't finish the coloring during the center time, so I have them complete all the problems first and then color....they can come back to the coloring if they finish an activity early on another day.)

I use this as a teacher-guided center sometimes, but I can also use the wheels as an independent center for review - students can copy the notes and complete the practice on their own. Coloring is similar to color by number - that part is last and they can return to it later. These get added to students' binders to keep for the year, so they can finish coloring any time they finish something early.

I make this a collaborative center (I love when students have math conversations!), but it could be independent work.

This is a fun activity to practice more general knowledge, with less calculation involved. Students can just quiz each other and share correct answers, or they can record their answers to be checked later. I have three sets so far - Numbers and Operations, Geometry, and Algebraic Concepts (link to download them is above.)

As you can see, some of the center rotations require students to work together, while others allow them to work independently. I like to have this mix, so students can share ideas and solving methods but also have time to work with the concepts and skills on their own. I also like to compare the work they did alone with the work they completed with others.

When students complete a center assignment, I have them put that assignment into a specific tray in my classroom. I go through the trays at the end of each day and use my checklist to record who's handed in their work.

I do grade all of the center activities....sometimes it takes me quite a while!!

When students complete a center assignment, I have them put that assignment into a specific tray in my classroom. I go through the trays at the end of each day and use my checklist to record who's handed in their work.

I do grade all of the center activities....sometimes it takes me quite a while!!

I have sets of 4-5 center activities for each of the topics below, so these are the ones I use/have used. I don't use all of them every year (because, time!). I pick and choose based on student needs, time of year, etc. If you're interested in trying any of them, click the titles to see them in my TPT store. (I have 2-3 activities for many other topics, but won't bundle those until I have 4-5.)

Absolute Value

Area and Perimeter

Fractions, Decimals, Percents

Mean, Median, Mode, Range

Order of Operations

Percent of a Number

Ratios and Proportions

]]>Absolute Value

Area and Perimeter

Fractions, Decimals, Percents

Mean, Median, Mode, Range

Order of Operations

Percent of a Number

Ratios and Proportions

For Teacher Appreciation week, I created two FREE problem solving math wheels (they are in the same PDF file) - they can be used to teach problem solving strategies, be used as a center activity, or be used as a finished early activity. When complete, they can be added to students' binders/interactive notebooks to be used as references all year.

I hope you can use them! Just click the image to download.

]]>I hope you can use them! Just click the image to download.