## Mastering the Basic Math Facts – Addition and Subtraction

** Objectives**

- To learn about strategies, activities, and interventions to help move students beyond memorization of math facts to mastery.
- To deepen our own understanding and enhance our professional toolkit in meeting the needs of our students whether it is to close the achievement gap or extend their conceptual understanding.
**To work as a vertically aligned professional learning community**(PLC) to create a more student-centered math classroom where students may select their own tools to investigate math concepts.<p>

**Requirements**

- We will answer two questions and then comment on at least 2 of the posts made by members of our PLC.
- In responding to posts, we will clearly state which aspect of our peer’s comments resonated with us and then add additional insights/examples either from the text or classroom experiences.
- If there are concepts that are unclear or examples discussed that we would like additional information about, questions may asked of members of the PLC.
- We will endeavor to post in a timely manner so we may maximize our learning time and capitalize on the expertise of members of our PLC.
*We will earn 16 professional development credits for the book study.*

**Introduction & Chapter 1**

*Answer two of the following questions and then respond to two posts made by members of our PLC. *

**W1-Q1)** Why is mastery of math facts important? What problems have you observed when student do not know basic math facts?

**W1-Q2)** What have you observed about anxiety related to memorizing math facts? Are there types of math facts practice activities that increase anxiety or decrease anxiety?

**W1-Q3)** How would you introduce the concept of math facts to your class and justify the need to know them?

**W1-Q4)** How might attention to the sequence in which facts are introduced support mastery of the facts.

**W1-Q5)** What real-world experiences might create an effective context for addition or subtraction problems?

**Week 2 **

**Chapter 2: Plus One and Plus Two**

*After reading pages 31-50, answer two of the following questions and then respond to two posts made by members of our PLC.*

**W2-Q1)** Reflect on the importance of students’ understanding of the commutative property. In what ways will it support their success?

**W2-Q2)** Reflect on the inverse relationship between addition and subtraction. How will this early understanding support students’ success as they continue their mathematical journey?

**W2-Q3)** “Subtraction is a separation or comparison process” (p.33). How could we utilize the model drawings provided by our district to help students grasp this concept?

**W2-Q4)** After reading pages 41- 50, select an activity/game that you could adopt or adapt to help students build automaticity. Explain how you would introduce it to the class and monitor students’ progress towards mastery.

**W2-Q5)** What are two literature books that you could use (or have used) to help students develop an understanding of +1/+2 or -1/-2? Provide the title and summary of the text and justify why they are appropriate choices.

**Week 3 **

**Adding Zero and Classroom Management**

*After reading pages 51-62, answer two of the following questions and then respond to two posts made by members of our PLC.*

**W3-Q1)** Adding and subtracting zero is an abstract concept and might be confusing for some students. What strategies might you use to help students build their understanding of adding and subtracting zero?

**W3-Q2)** On page 52, there are four “big ideas” presented that will help with our exploration of adding and subtracting zero. Select one of the big ideas and suggest a hand-on or kinesthetic activity that you could use to help support that idea.

**W3-Q3)** On our campus we are creating student-centered math classrooms in which students are able to self-select the tools they find most useful in helping them to grasp concepts. What routines and procedures will you need to have in place to help students self-select their tools?

**W3-Q4)** How will you manage math games in your classroom? How will you keep track of students who are experiencing challenges?

Q1) Learning math facts with automaticity is a critical component of solving more complex mathematical problems that I realized. Brain research shows that students who have difficulty recalling facts actually focus on solving the actual fact and not on the other aspects of the math problem, especially at upper levels when dealing with decimals or fractions. By learning their facts and being able to recall them with ease, they are able to analyze and think more about the complexities of math problems.

Q4) Introducing facts in the order suggested by the book seems very important to me because student will learn over 80 of the basic facts out of 121 total facts. Adding 1 and 2, friends of 10 (which they learned in 1st grade), doubles and sums of 10 consist of very few steps and the payoff is huge! It also allows students to think more flexibly because once these facts are mastered, they will be able to use these facts to think about addition/subtraction facts differently – 8+4 becomes 10+2 which is easier to solve. Since our number system is base 10, it seems to me that this is something that they will be able to use with little difficulty once they understand the concept fully. The same can be said of doubles – if you know 3+3 =6 then you should be able to easily grasp that 3+4 is going to be one more, or 7.

Mary Ellen you stated, “Adding 1 and 2, friends of 10, doubles, and sums of 10 consist of very few steps and the payoff is huge!” As I thought about that statement, I looked back on page 12 of our text and the sequence suggested for teaching math facts. Integrating fact fluency into the curriculum is necessary for us to help our students become proficient. So many times we point out that our students do not know their facts but we rush through the teaching cycle for facts mastery, or having explored the concept we moved on before the students fully grasped it. Kinder spend a lot of time building the foundation of our students’ number sense, but if it is not reinforced beyond that grade level the students will not master it.

Earlier today I was talking with a fourth grade teacher from another school district and she shared that one of her students could not find the page in a book when he was asked to find it. He could not count. He did not understand the sequence of counting and the fact that the number was between 100 and 200. If a fourth grader is struggling to count up to 200, what does that say about our education system? If he is struggling with basic facts, are we realistic in expecting him to pass state assessments? What will the continued struggles in math and in school mean as he moves beyond elementary grades?

You said research states that “students who have difficulty recalling facts actually focus on solving the actual fact and not on the other aspects of the math problem”. I totally saw both sides of this in my classroom last year! My students who were able to recall composing and decomposing seemed to have it all figured out when we got to 1 and 2 more as well as 1 and 2 less. It was amazing to see what they could do when we got to addition and subtraction!

Mary Ellen, your first comment “learning math facts with automaticity is a critical component of solving more complex mathematical problems than I realized’ was most evident in my fifth grade classroom last year. Some of my students (G.T., “average”, and “below” grade level) did not have automatic recall of simple addition and subtraction and would count on their fingers. (Many of these students would do so under the cubby of their table as if embarrassed by the situation.) This deficit caused students to labor over the facts which limited their ability to “analyze and think more about the complexities of math problems.” What should have taken 3-5 minutes to solve a word problem took 7, 10, or more minutes to complete. This typically frustrated those students because they knew that more problems were needed for the assignment.

What real-world experiences might create an effective context for addition or subtraction problems?

There are a lot of concrete examples to make the abstract concepts of addition and subtraction clearer for younger students.

Food – if we want more food then we are asking for food to be added to our plates. When we go to the grocery store and we buy groceries, the amount of food we have in our homes will increase.

Think about Halloween, children may collect a lot of candy (add to their total), but their parents may separate the candies and they are given a piece at a time.

Friends/Games – there are some games that can be played individually, but oftentimes we want our friends to join us in playing the games. (It is often helpful to act out this scenario). We may even talk about combining points in a game, and play a quick game to reinforce the concept that when we combine the points we oftentimes have more.

If friends come over to play a game, later they will leave to go home. (Addition and subtraction)

Money – When we use money we have less than the amount we started off with. Buying and selling objects is a great center for early elementary students to explore to see how addition and subtraction is used.

Stickers – at school students may earn stickers. They may put together the stickers earned throughout the week/month to see how many stickers they earned and which prize they may be able to “purchase” with their stickers.

Academic Bowl – from Kinder to 5th grade, students are now able to determine who the winner of the round is based on the group who earned the most points.

Time – some teachers may use a digital clock to show students how much time is left to complete an activity.

Some students may need to practice a skill during recess that they struggle with, and this time is taken out of their overall recess time.

Regardless of what we do, we cannot separate the need for addition and subtraction from our daily lives.

Sherry J

I love how easy it is to see how many ways and places you use addition and subtraction in your daily lives! Making it applicable to a student is the best way for them to learn and the experiences you stated, every Redhawk experiences almost all of them!

We purchase with our Redhawks tickets they earn and it’s been a very interesting study in more than/less than. They have to figure out if they have enough, if not, how many more do they need and if they have more than they need, how many do they get to keep. We did it weekly and it was a great learning experience! I might have them actual record some of the numbers in the Spring next year.

Michelle, I like the way you use the ticket idea to reinforce the concept of more and less. Another way to extend the activity with the ticket is to use a ten frame to help the students build the numbers. If we use weighted number then they could actually SEE the numbers that are more and less. Identifying the number on a number line would also allow for movement and reinforcement of the concept.

Q1) I know I have heard over and over again how important it is for students to learn basic math facts such as 1 more, 2 more, 1 less, 2 less, and friends of 10. When I look at the sequencing guide I see the topics that we spend so much time on in kindergarten right at the top. I also see the challenges students will face if they don’t grasp them at an early age because math builds on itself. Without the automaticity we so desperately want all of our students to have, they will struggle to focus on other math processes. When students don’t know basic math facts (such as counting to 20 or even 10 in kinder), it is very difficult for them to figure out 1 more.

Q3) When I begin teaching basic facts such as 1 more, I always start with real life examples and use my students to make it as concrete as possible. I can show them how they use these facts in everyday life because the stories we tell are real. By giving them the opportunity to see, hear, and experience using these math facts, they begin to know how important they are to us.

Ashlee – I thought the same thing when I saw the sequence. I was so frustrated this year spending so much time on 1 more and 1 less and then 2 more and 2 less, but it’s so important to building their foundation. Now I feel like we’ll have more ideas for teaching and practicing it next year and maybe I won’t be so frustrated 😉

Ashlee and Michelle, I look forward to hearing some of the ideas that you now have regarding teaching more and less to our students. I know regardless of what strategies we use, manipulative will be a huge part of the decision.

Ashlee, you talked about the importance of learning various concepts including the friends of ten. Since we have a base ten system, it cannot be adequately stated how important developing that conceptual understanding is to students’ success. I know revisiting the same concept repeatedly sometimes get boring for teachers, but for the students since it is really their first exposure it is not as boring as it may appear to us. Nonetheless, when we see students become disengaged, the onus rests upon our shoulders to adapt the instructional strategies to make the concept more engaging.

Q2) When I taught older grades (4th and 5th grade math) I saw a lot of anxiety associated with memorizing math facts. It almost became a mental block for some kids. By the time they got to me they had not memorized them, and then every time they had to think through the sum or difference and it took them longer and they would get more and more frustrated. Working the problems took twice as long, which fed into their belief that they “couldn’t do math”. When in reality it wasn’t that they couldn’t DO it, it was more that they took longer and became more frustrated. Their peers had all finished and they, of course, realized that and started to give up. Just like it stated in the book, any type of “competition” or “speed” game was crushing for those kids – the ones who needed the practice the most! (I thought that was a great point in the book, about the games where kids who didn’t have automaticity would be “out” first when they were the kids who needed the practice more than anyone. Pg.6)

Q3) For me the easiest way to begin to introduce math facts to Kindergarten is through literature. I appreciate that this book references “math” books that can be used to introduce concepts. In reality you can use almost any book to talk about some type of math concept, with a little creativity 🙂 I think by introducing math facts through literature it’s a natural transition for them, because we read constantly. It’s also a great way to review because we often read our “favorites” over and over. And if we don’t talk about the math concept in it the first few times we read it, how great to “discover” the math in it when we read it again – a way of looking for new things in the book (or pictures). Also, as it mentioned in the book (p.26) there is math all around us, all day. Just talking through how many boys, how many girls, how many more, you have 8 books and I need you to have ten, how many more do you need to choose….makes it very real to them when we finally introduce actual math facts.

Michelle you pointed out that students become frustrated when they do not know math facts. One of the activities that I do not like is timed math fact quiz. This idea of “beat the clock” does not instill confidence in students already struggling with the concept. Whether it is addition, subtraction, multiplication, or division, students who struggle with math facts will not last.

How can we include competition into mastering math facts? I offer, instead of having individual competition, we could use a team approach. Students get the opportunity to discuss their answers prior to answering. Speed is not the goal with this approach but rather mastery through conversation, manipulative, and practice.

Thoughts anyone?

Michelle,

Although I’m new to the math teaching game, I have witnessed my own daughter feel these same frustrations when solving problems. Until fourth grade, she was very resistant to learning her math facts. That made her take longer on assignments, creating more and more anxiety. Once she memorized them, her work came easier and I noticed she was less stressed.

It’s hard to see their frustration when they are solving problems slower than some of their peers. Back in the day when you gave Thursday timed math fact quizzes, some students beamed and some slumped. It was boring and redundant. I love that we can assess whether they know the facts in so many other ways. We can change them up to keep their interest.

Sherry, you have included a lengthy list of real-life experiences for addition and subtraction.

Over several days at the end of the school year, the fifth grade students listened to engineers share about their jobs, whether it be mechanical, electrical, chemical, etc. One statement that each engineer consistently gave was the need for understanding math. Some students who seemed interested in being an engineer groaned. For me, these talks affirmed the importance of what had been shared throughout the year: math is everywhere. The students were given an opportunity to build a home or city as their final math project. Their creativity was obvious; many partnerships had “out of the box” results. However, when it came to the actual math solving portion (perimeter, area, and volume), some were less inclined to embrace the project. Sure, multiplication was needed, but so was the adding of the products of the two lengths and two widths. Even determining the length or width proved challenging when it involved a fraction.

Leslie, the challenge to make math meaningful continues to be one that mathematicians have contend with over the years as we attempt to help students realize the relevance of what they are learning in the classroom. Despite the pace at which we are having success, we should endeavor to help students realize that math is integral to their daily lives. Simple activities such as walking around the school could help reinforce the idea of perimeter. Walking across the hard top could help reinforce the idea of area and perimeter. Having conversations about eating pizza or a cookie, could help with their understanding of fraction. When we allow our minds to engage in creative thoughts, we too will be amazed at how many ideas will emerge that will help us meet the needs of our students.

Michelle,

I loved the way you talked about kids “discovering” the math within the literature. By discovering this on their own, they begin to see those real life connections and the relevance of math in their own lives. As Sherry so eloquently stated, “We cannot separate the need for addition and subtraction from our daily needs.” This is so true. We need to allow our students to discover the real uses of math and where they are seen each day. We can use addition/subtraction when they have to figure out how many more stickers they need to cash in for behavior incentives, to answer the unending list of questions “how many more minutes until…how much more money do I need, how”, and so many other opportunities in the classroom. Instead of answering this question for them, allowing them to figure it out and deciding how to get from point A to point B helps them make that critical connection of NEED.

Leslie,

I also agree with your comment about how focusing on the solving of addition/subtraction is very frustrating for students, especially those older kids that experience embarrassment because they don’t know their facts from first and second grade. They make assumptions that if they cannot solve something as easy as adding without laboring over it, then they certainly cannot understand those higher level questions and solve those. This breeds the thinking by most that “math is hard” and “I’m not good in math”. I will definitely make this a top priority in my math classes this year…even more so in years past.

W1 Q1) According to research as well as personal experience teaching math, automaticity of basic math facts frees the brain to focus on other math processes. If the basic math facts are already memorized then the speed of completing complicated math tasks will improve. As the students progress through the elementary years the math tasks get more complex, if they already know their basic addition and subtraction facts then it relieves the brain off of computing and thereby focus more on the math process. If the students are unable to develop automatic recall of addition and subtraction facts then they experience frustrations while doing more complicated concepts like multiplication (repeated addition), division (repeated subtraction), fractions, decimals, and then algebra. They spend more time doing simple addition/ subtraction tasks than focusing on the new concept.

W2 Q2) Several students do experience anxiety while memorizing math facts. They seem to squirm when asked 5+2 as they want to do it as quickly as possible or faster than a fellow student. Hence they should be asked to compete with no one but themselves. eg. See how many facts they can complete in a minute and then make a note of it, so that next time they can beat their own score. Such students can get very anxious if asked to complete all facts in 1 minute with the rest of the class. However if they are asked to do the same activity, but instead focus on the number they can complete in 1 minute then they feel more successful and their anxiety will decrease.

Shradha, In response to your comments on W2Q2, I noticed that you said several students experience anxiety while memorizing math facts. I thought that was interesting because just like everything we try to teach students there is not one 100% correct method to reach every student. So while, some students thrive on the competition with themselves when completing timed math quizzes, others have extreme anxiety in the same situation. We find this throughout our lives, while some people are comfortable or enjoy public speaking, I would have a huge anxiety attack, if asked to speak in front of people older than 11. I think it goes back to knowing that if students don’t learn the way we teach, then we teach the way they learn.

Shradha,

I completely agree that kids should not be competing or comparing themselves with other students. By focusing on their own performance, we are emphasizing growth mindset and individual improvement.

Hi all! As I’ve been reading I keep having flashbacks to my childhood and learning multiplication facts. My mom would sit me at the table with a sheet of 25 multiplication problems, then she would put a glass of water in the microwave and set the cook time for one minute as a timer. I remember doing this repeatedly until I knew them without hesitation. I realized a couple of things. I learned all of the required math facts, but I learned them in isolation and without seeing the connection to and flexibility within our number system. In chapter one, focusing on understanding the number system before building fluency, gave me a better understanding of how to set the foundation for students so that they become fluid in thinking about numbers before focusing on facts. This was also interesting to me because when I taught fifth grade, I was so shocked at the number of children that didn’t know their math facts. Therefore, this made learning new concepts extremely laborious and difficult. Helping children become confident in their ability to manipulate numbers will help avoid their blocks to math facts and decrease their anxiety when having to recall basic facts

Ms. Johnson, I completely agree with you on not liking the “beat the clock” math quiz. Such a quiz only makes the quiz taker more anxious and defeated. Even if he/ she knows all his concepts with automaticity he can make mistakes because of the race . Instead of beating the clock the students should try to beat their own previous score . This will give them more confidence as they improve in getting more facts solved within the time frame as it is rightly said ” It doesn’t get easier, you just get better” .

Ms. Truncali, I loved the progression of developing math facts. The friends of 10 and doubles just simply helps them understand the math patterns so much better. If the students get an understanding of these patterns, then memorizing these math facts will become so much easier for them as they shall start seeing these patterns and hence 8+4 will take them automatically to 10+2. in the same way it will be super easy for them to add multiples of 10 like 2+7 =9 and 20+70=90 .

Q3) Helping children understand and explore the big ideas of math is crucial in developing their mathematical thinking. Understanding the commutative property, that addition and subtraction are inverse operations and that numbers are flexible will lay the foundation for learning math facts. The use of number lines, models, manipulatives and partners throughout math lessons will help our students be able to understand and visualize the patterns in our number system. I think it also important that while learning these big ideas, they are also writing explanations of their thinking. Critical writing will help them make sense of what they are learning.

I agree that understanding the communicative process benefits students when they are learning their math facts. If we’re using flash cards, they should see both 1+3 and 3+1. Plus younger elementary kids love to say “communicative”. It’s 5 claps and 13 letters!!!

W2-Q1 – Understanding the commutative property dramatically decreases the number of math facts that a person learns. Therefore, when learning math facts, the overall goal is not as daunting or intimidating. I think this will definitely relieve some of the anxiety that students face when learning math facts. Most importantly, the commutative property shows the flexibility within our number system which help develop a more fluid understanding of numbers.

W2-Q2 -Teaching students that addition and subtraction are inverse processes allows students to understand another pattern within our number system, helps develop fluency with numbers and automaticity with addition and subtraction facts. Helping students learn that joining and separating and adding and subtracting are all related will allow them to generate and solve problems with greater ease.

Jodi – when we were up at school planning for math next year, one of the big units was “patterns and functions”. I kept thinking, how can we get 20 something days out of patterns!? But I think if show the patterns of addition and subtraction it’s a good way to apply patterns to a different part of math instead of just ABABAB.

Ms Ng. I completely agree if the students are able to understand addition and subtraction as an inverse process it will make them so excited that they already know the difference by simply looking at the addition sentence. This will further improve the automaticity in their brain as they are looking at addition and subtraction at the same time. Also the joining and separating part will ease them understand the part-part -whole concept even better and put all the fact families under the same umbrella.

Shradha and Jodi,

How do you think understanding the relationship between addition and subtraction will help students grasp the concept of multiplication and division?

I agree that it’s beneficial for them to understand that addition and subtraction are an inverse process. With newer learners, telling stories with pictures, student helpers, manipulatives help. You begin with adding things together and then you go in “reverse” as a ways of revisiting the story. Example: 4 fish were swimming in the lake. 3 turtles jumped in and began to swim with the fish. How many fish were swimming? The 3 turtles got out of the water. How many are swimming now? “That’s the same as when we started the story!” 4+3=7, then 7-3=4.

Sherry, when you stated that you do not like time math fact quizzes, I found that interesting for several reasons. Being a product of timed math quizzes, I always felt that my ability to recall facts with speed and accuracy was crucial in helping me learn new concepts that were being taught. It was and is also extremely beneficial in everyday, real life situations. I would have been and would still be lost in many situations if I didn’t have automatic recall of these facts. However, as I thought more about your statement, I kept going back to the fact that as I learned math facts and took timed math quizzes, these were always in isolation and never related back to any mathematical concept or understanding. Understanding the patterns within our mathematical system and how to manipulate numbers as a foundation to learning math facts will help students to continually develop their understanding of mathematical relationships.

Michelle, in response to your post about using literature to introduce math concepts, I completely agree. Children’s literature is a love of mine! Being able to provide opportunities for children to make connections between literature and concepts they are learning is powerful.

W2-Q1

The commutative property seems a critical understanding for increasing automaticity with addition math facts. First of all, students, once they understand that adding is a combining of objects in a set number of groups, need to grasp the concept that order does not matter in addition. They need to deeply understand that 4+2 and 2+4 is simply combining the same number in each group in a different order. Young children often see these related facts as two separate and entirely different equations – totally unrelated. However, if they can use the commutative property with ease, they begin to understand, on a deeper level, the relationship between numbers and the joining of those groups through addition. As a bonus, they will only have to learn half of the addition facts versus all. Using the commutative property also helps build the concept of fluidity of thinking when students make this connection.

W2-Q2 Understanding inverse operations also deepens the understanding of number relationships and connects the addition facts to the subtraction facts they are memorizing. For me, it is so much easier to add in my head than to subtract. Even when I was a student, I remember thinking about subtraction in relationship to addition. Because I knew my addition facts so well, I could easily transition to subtraction because I understood the relationship between addition and subtraction. I often used this type of thinking with my students last year because those that had a strong grasp of inverse operations, were able to learn and recall their subtraction facts much more quickly.

Mary Ellen – after reading your answer to the first question I think I’m going to try something different when we do simple addition at the end of the year. If I have the children hold the numbers and make a “human” equation and then move them around to show that it doesn’t matter who comes first and who comes second, the sum doesn’t change. I hope it will be a more visual way for children to see the commutative property.

Jodi, I appreciate your insight that learning the commutative property will help alleviate anxiety for some students. It is so critical that students realize that our number system, including add/sub facts, are made of patterns that are predictable. That alone, makes math much less daunting to kids as the majority find patterns to be fairly easy to learn. Once kids can approach math in a relaxed and confident manner, it will help them realize that they really are mathematicians!

Sherry, I agree with you about timed tests. I also like what Shradha said about having students working to beat their own times. Next year, I hope to have the kids use the monitoring checks provided by the author and have them keep up with their progress by creating a progress chart for them to keep in their math folders. This way, students can see their progress in a private setting and I can use it during conferences or to reinforce those facts with which they are having difficulty.

our PLC.

W1-Q1)

Basic math facts are of significant importance in order for students to not only complete more complex problems, but also so they can do it with ease and little stress. Although I’ve never taught math before, I have taught music note values, which is math. When we would talk about measures and the number of beats each note gets, I could easily tell the difference in those that knew basic math facts and those that didn’t. For the ones that didn’t, the question became multi-step and frustrated them quickly. However, for the others, solving the number of beat question was simply recall of values. I anticipate the problem in my math class being that students do not finish work in time, resulting in frustration, worry or anxiety. Even though we tell them not to compare themselves to others, it will happen. My goal will be to stress individual improvement and success opposed to competition amongst their classmates.

W1-Q2)

I have helped many if classes work on basic math facts in music. Most of the time, the teachers gave me flashcards to drill them on in line before they were picked up. The problem we would have with this is that some students were very quick at recalling the answers and they would shout them out. The slower students eventually stoppe trying. We made it work better by working with them individually down the line instead of the full class.

When I was in elementary school, I remember taking timed tests. After we finished, we would check them as a class. Then, the teacher would rank us in order of how many we got correct. This was a HUGE stressor for me and others. Instead of building basic math facts fluency, we were learning to nervously complete as fast as possible. We didn’t know why the answers were right or wrong, meaning little to no number sense. Instead of using the timed tests in order to compare, I would prefer to use it as a personal evaluative tool. Activities with little to no risk helps to decrease anxiety and improve performance. I feel like games and fun, low risk activities are a great way to help some students.

I hope to use these experiences to better educate the students in my classroom. The foundations we lay in the primary grades will help these students to build stamina and competency in math.

Kristi,

We concur with the statements you made regarding the stress relating to timed tests and the fact that the students who need the practice do not really get a chance to practice the skill that they desperately need. The challenge we now face is… are we going to repeat the mistakes of our teachers or are we going to to do what we know is best for students? We are no longer trapped by traditions. With the research that we have available, we can now make informed decisions regarding our instructional practices and focus on developing fluency in math instead of rote memorization.

We remember the extremes of the instructional continuum. We will not forget the engaging and interesting classes, and we will not forget the boring classes that made us want to sleep. I always want to have a class that is highly engaged, meaningful, and memorable. It takes hard work, but it is worth every extra minute to see the smile on a kid’s face at the end of the day.

Kristi,

I love the example of math used in music! This will be helpful for your students to see a real world example and help them understand the importance of fact mastery.

W2 – Q4 – I like the “Hop the Line” game because it can involve both addition and subtraction, so after you have introduced or taught it the first time, you only have to revisit to use it for subtraction. I would introduce it by showing the number line (which I will already have in my room) and showing how to draw the card, the spin the spinner for either +/- 1 or 2. I think it’s important to demonstrate to the whole class what “hopping” looks like in our room – as opposed to running, jumping, etc….Then show them how to record their answer. This activity is great for Kindergarten as it gives them a way to kinesthetically be involved in their learning. I would monitor their progress by checking their record sheets, and also I would have the game in my rotations so that after they have played it a few times I would put it as the “teacher” station to observe their understanding as well as their recording of the math fact.

W2 – Q5 – Two books I have used in my class are (they are both Stuart J Murphy books). “Monster Musical Chairs” – which the kids love the repetitive rhyme of it and they quickly pick up on the pattern of one less each page as most have played musical chairs in their lives. The basis of the story is the they are monsters at a party playing musical chairs and each round one more is gone. You can predict how many are left and the children love predicting (the first time you read) which color monster will be gone after each round. It’s a fun way to introduce that game as well. And “Jack the Builder” – which introduces the concept of not only +/-1 or 2, but larger numbers as well. This book I usually use towards the end of the year and I like that it incorporates review of shapes into our learning as well. It is a story about a boy building with shape blocks, but then for each thing he builds his imagination sees it as something much grander!

Ms. Bellomy, I would definitely love to read the books that you mentioned. Monster Musical Chairs and Jack the builder sound like fun books and reading these to them after a break would help them bring back those facts that are connected with the book. Literature helps them make more sense of the number facts in a different perspective. Also rhymes help them remember facts better.

Michelle,

The Stuart J Murphy books are great! He has a bunch that touch on other math concepts too.

Kelly,

What are some of the Stuart J Murphy books that you have found useful? How did you incorporate those books into the math lesson?

Michelle,

I also liked the Hop the Line activity! It is one that has so many possibilities for teaching, plus it involves them being physically involved in the process. I think it would be a very useful activity for Kinder or really any grade. A lot of the older kids that have a hard time grasping the concepts would most likely benefit from physically moving their bodies. I’ve done similar activities in music and it definitely solidifies a concept that can be abstract and strange to some students.

It is so hard for younger kids to sit still. Any lessons that include a kinesthetic component will benefit their learning and their abilities to stay focused. “Hop the Line” will keep them moving when they get to do this in small groups. This often helps them stay on task. For some, using colored bears on a smaller line can be easier.

W2-Q4) Introducing math games to develop automaticity is indeed a part of the day that students look forward to. I love the Fact card jumps that can be played on the recording sheet or on a floor- sized number line using painter’s tape. This would also incorporate mobility for the students especially in the morning and help them wake up. This can be done with the +1/ +2 cards or -1/ -2 cards to help the kids add and subtract. I would model it myself to introduce the activity and to show how the game is played. After a week of practicing a certain fact I will have them do a fact check where they record the correct number of questions they completed in 1 minute. They shall record this down and next time try to beat their own score. This can also provide as an informal assessment that can easily be used when need arises.

Hop the line or the Fact card decks are also great ideas to help the kids develop automaticity in a fun way.

Shradha,

I think you are right that this game would help with automaticity. It isn’t directly timing them on how many they can get right in x amount of time. However, they will inherently learn how to answer problems quickly after repeatedly doing them during the game. I may need to work with you to implement this in my class as well.

W2- Q5) Mouse Count by Ellen Walsh and Counting Crocodiles by Judy Sierra are 2 books that I would like to use in my class to further develop students understanding of adding +1/+2 facts and subtracting -1/-2 facts. The Mouse count is a book of 10 clever mice that escape from a greedy snake that wants to eat them for dinner. This can be done with the Mice in a Jar game. Where they will have bags with 1-10 mice cut out. They pretend to be snakes and they reach out to a bag and count the mice in there. Then they put this down on a recording sheet. They add 1 more and 2 more to the above number. Doing this activity with literature provides an engaging context for the study of math facts.

Counting Crocodiles is another counting book that includes counting 1-10 as well as 10-1 as a clever monkey counts on crocodiles on his travels from an island with lemons to an island with bananas. As you read along the first part ask them to predict what 1 more would be. In the same way ask them in the second part what 1 less would be. They can also record their answers using equations. You can read the book several times.

Students love books about animals. This could be taken further by asking them to write their own story problems involving animals.

I have not read these books but I am going to go get them! I love the ideas and I love that they are interactive.

Shradha, I spent the day reading math books to add to my toolbox. Counting Crocodiles was one of the books I read (online). I love the fact that we can now readily access math books online and even have the option of read aloud. No longer are we trapped with not having book at our disposal. As we continue to add the literature component to our math curriculum we are making math more meaningful. Integrating math with across the curriculum will help to show the relevance of reading which helps to reinforce the process skills that students are required to apply.

W2-Q1- It is extremely important for students to understand the commutative property. Once students recognize that they can do this, memorizing their facts becomes easier. Which in turn will build their confidence and from my experience, once they begin to feel confident in Math they tend to try harder. They feel their success and they want more.

W2-Q4- I really like the Hop the Line game. Not only does this game get students up and moving(which we love) it also keeps them engaged. I would introduce this game by setting it up in my classroom and choosing a student to play a round or two with me. Then I would choose two more students and let them play a few rounds for their classmates. Their are many things that I love about this game but one main thing is that the partners have to work together. I noticed a few times last year in my math rotations that students would often try to work independently when my goal for the specific activity was to get them to communicate a work together. With this game, one partner is actually acting it out while the other partner is telling them what to do. I also love that the students are keeping track of their progress. It holds them accountable, but it also allows me to come around and see how they are doing as I am also working with other groups.

Melissa, I love the fact that you are encouraging student discourse in the math workshop and now you have shared additional ideas how to hold them accountable. One of the things that I think you may find helpful in encouraging dialogue is the Math Congress. During this time, encourage students to share their thoughts about the strategies they used independently and with their partners and although all of the students may not do it at first I have found that when I praise and recognize the students who are following through, then more students tend to follow the directions given or modeled. I like the fact that you are focusing on holding the students accountable for their work and for keeping track of their progress.

Melissa,

I too love games that allow students to work together and talk about their learning. When students work together they are able to help each other where one might be struggling.

W3- Q2) On pg. 52 the 2 big ideas that if one addend is 0 then nothing is being added or joined, and if nothing is being separated from a set the quantity in the set remains the same. This is a very important idea for them to understand and this can be done also by giving them real life examples and helping them write them into number sentences.

I liked the Fact Family homes activity on pg. 61 where each partner picks up a card and writes that number along with 0 on the door of the Fact Family house. They then will write 4 fact family number sentences in each of the windows of the house. I liked how this activity incorporates addition and subtraction with 0, and also emphasizes the commutative property of addition and subtraction.

W3-Q4) In order to have the math games working smoothly in the classrooms the students need to know the teacher’s expectations and rules of the game. These expectation sneed to be told them several times in the beginning then reminding them several times while they are at it. This can be taught to them through Role playing and by displaying a chart with pictures near the centers where they shall be playing the math games. It is very important that this is done in a quiet manner and the students stay at their acceptable voice level as you could be working at this time with students who need more attention with certain concepts You can guide them through the games or play it with them till they know what is to be done. The students also need to know that if they are unable to follow the rules they might lose their privilege of playing the math games and will instead do math fact practice tasks independently on the seats.

I also use pictures when using tubs, math games, and small group activities. I try to give step by step instructions using simple pictures. If they first need to roll a dice, after the one is a picture of the dice. (1. Dice) I always include “next player” with a picture that is consistent throughout all games and explicit cleaning directions. Example: Put cards in the bag, put the dice in the container, put the lid on the box, etc. I do my best to find pictures that go with cleaning.

On the bottom or back of the directions page, I put a volume icon that stays consistent all year. This is visited all throughout the year and a chart (with the same icons) is visible in the room.

W3-Q4

I agree with Shradha when she says must be in place. As I go about setting up my classroom I plan to explicitly teach the expectations each day for multiple weeks – as long as it takes for the students to fully master them. I believe, however, that not only should these expectations be taught and modeled, students must also see examples of what is NOT acceptable in group time. By allowing students to model the expectations correctly and then incorrectly, they can actually see the behaviors that may get them off task or in trouble. They will be allowed to use student discourse to explain to others in “kidspeak” and it allows them to review the rules for themselves as well.

I am planning to have each station color coded and all materials kept in some type of tote with a lid. There will be a captain assigned to each group. That captain is responsible for making sure materials are placed in the appropriate tote and in the appropriate way at the end of each session.

I feel strongly that students should have something to turn in or record their information on at each session. This helps students maintain on task behavior and gives them some accountability to make sure they are working instead of just going through the motions or playing.

There will be some type of switch schedule so that each child knows where they are to go without teacher direction. It should be something that gives me an easy way to move the kids and makes it very simple for kids to understand the order in which they should move through the stations.

I also use a poster in my classroom that allows students a visual of voice level. Since we will be doing several groups, I plan to make one for each station so students will have a visual at each place they visit.

W3-Q1

Adding and subtracting zero is something I think we, as adults, think is very simple for students to understand. Therefore, we may not spend an adequate amount of time on this concept to ensure students understand fully what zero means. I believe students need lots of opportunities to practice using zero in addition and subtraction. They need to use manipulatives of various types and they need to be presented with many different scenarios so that they see examples of real-life uses of zero. Students should be presented with a problem within the classroom, then act it out with real materials; pencils, markers, paper clips, etc…Then, allow students to discuss what happened when you add or subtract zero and connect that discourse to the symbolic structure of an equation.

As students become more efficient, they can then begin to write some of their own story problems using manipulatives. They will tell the story, then allow other students to solve their problems or present them each day to solve as a whole class. Student discourse is key.

They can also listen to and sing songs about adding and subtracting zero.

I agree Mary Ellen. I always think, “How long could this possibly take it’s “0”! Just say the other addend!” And while kids might memorize it quickly, it takes a lot of practice and activities to understand the CONCEPT of adding “0”. I think this chapter brought to light for me that I need to spend more time focusing on a true understanding of the concept.

W3 – Q1: I think in Kindergarten the most effective way for children to understand most concepts is kinesthetically in the beginning. It mentioned in the book “acting out” the concept and I think that would be the best way to begin to build their understanding. I would start by demonstrating with 3 kids at the front and ask if we added “0” kids to this group, how many kids would we have. I think seeing that when you add “0” there is nothing added to your number, visually, with other children it will stick with them more.

W3- Q2: I think the “big idea” that is easiest to demonstrate kinesthetically is the commutative property. You can have children choose a number and hold it up, and then choose an addend (from a selection that you have pre-selected). They can stand close to each other and there can be a child in the middle making the “+” sign with his/her arms. Then you can work together as a class to figure out the sum, have a child hold out both arms as the “=” sign and a fifth child to hold up the sum. They can write it themselves on a card after you have solved the problem together as a class. Then have the children not standing up close and cover their eyes for a “surprise”. While their eyes are closed, switch the two addends. When they open their eyes ask them what has changed. Ask if they will need to change the sum. Then solve the problem again to see that the sum has stayed the same.

I agree that kinesthetic learning at the beginning is more beneficial for kinders. Moving keeps them focused and excited about learning. MOST OF THE TIME, the students make better behavior choices when they are actively moving, rather than when sitting quietly while the teacher is talking.

Sorry I’m a little late to the party, I’m going to have to catch up! 😊

W1-Q1: mastery of math facts is important for students because it helps students calculate more quickly, efficiently and accurately. When students do not show mastery of math facts they tend to make more mistakes because they are relying on counting on their fingers or drawing lines etc. which leads to more mistakes because they tend to miscount. It also slows them down so they are not able to complete as many problems as their peers who have mastered their math facts. I’ve also observed students with poor mastery have less confidence when they approach a math problem.

W1-Q2: I’ve noticed that when students are provided with fun games and activities that help with mastering facts they tend to have less anxiety over the process. Students who just try to memorize without having a concrete activity tend to be more anxious and less successful with mastering their facts. Many of my students have used flash cards with a partner to play a war-type game with their facts. Whoever can call out the answer the fastest is the winner.

W2 -Q1: The commutative property is a constant conversation in my classroom. Whenever we are working on problems as a class I always ask my students to help me write the problem at least two different ways for example 5+3=8 and 3+5=8. Students will naturally write the problem out as it is presented in the question but often times the correct answer choice has the algorithm listed by its commutative property. Without constant dialogue about how the order of the addend does not change the sum students can easily make mistakes when selecting answer choices. I see this a lot with multiplication algorithms.

W2-Q2: Students who are able to make the connection that addition and subtraction are inverse processes will have a much easier time understanding that multiplication and division are also inverse processes. By understanding inverse processes students don’t just know two facts but actually 4 facts. Understanding this concept is crucial for students to be able to solve problems with ease and less anxiety.

W3Q4: Managing Stations/Tubs/Activities

I have always found having labeled containers with the topics and activities inside easiest to handle. After introducing the games whether whole /small groups then they were placed in their designated spots and containers and students could easily access them. Some of the games had a recording sheet that students turned in to me at the end of the activity (which varied from one day to multiple days), while others had a self-check sheet and the “graded” answer sheets were turned in to me.

Two of the challenges had always been how many activities to complete in one week and how often should they be changed. Starting with activities that are sustainable worked better for me. By sustainable, I mean activities where the concepts were reinforced but the manipulatives could be easily switched to make the game remain interesting. In so doing, I spent a longer time creating the new experiences, but the switch times were fast.

I struggle with finding activities that can last multiple days, up to a week. Last year I changed them every day, but I don’t know that I can keep up that kind of pace this year! It’s been good to find activities in this book that allow for the same activity, but different cards or numbers. This will help a lot with the prep for my math stations.

Hello Michelle,

Changing activities daily keeps the stations novel, but there are times when students may enjoy being able to continue deepening their thinking about a particular concept by exploring the concept through multiple manipulative. This year, I am hoping that we will continue providing students with experiences that will allow them to engage in using multiple tools so they will know which tools work better for them. In addition, I am hoping that we will go for depth and rigor so that they will boost their confidence as they move vertically.

I have been hearing a lot about the Toothy kits on TPT. These are basically games with a recording sheet being the laminated game mat. Have you all heard of these? I think it would be a fun and novel way of monitoring progress without it seeming like I’m taking a grade on everything.

I was actually wondering how often you all changed stations in the primary grades. It seems like they would enjoy the change, but sometimes they may want to repeat it. When they find an activity they like and learn from, do you repeat the center typically or wait a while to introduce it again?

W3Q1: Zero

If there is one concept about zero I want students to grasp by the end of the year it is… Zero is important! I think we should make it as concrete as possible. For example, have containers with objects inside of them and have students count, discuss, and record the number of objects are inside the containers. Give students number cards for them to select “treats” and encourage them to count the number of items they have. Zero should be one of the numbers included. When students get zero it should be discussed as the starting point. If student say it means they have nothing, encourage them to add “yet” to their comment.

W2Q3: Model Drawings

I love the posters that are provided by the district because they offer continuity from one grade to the next and from one campus to the other. When the model drawings are introduced to the class, I think multiple examples should be provided to allow students time to grasp what they mean and for the posters to become more than just posters that are placed in math classrooms. How can we help students to internalize the meaning of the posters? They must be constantly referenced, not just by the teacher but by the students as well. I offer that the students should be given the opportunity to pair their own drawings with the model drawings and explain their thinking about their pictures.

W2Q2: Inverse Relationship

The understanding of inverse relationship is a foundational skill that students need to grasp in order to progress beyond mere repetition and rote memory. The vocabulary words may be too advanced for lower elementary ( a subjective view, because it is not a TEK vocabulary) but they can understand the concept and later in life learn the term. The idea of experience before label applies… If students understand that 2+3=5, 3+2=5 and 5-2=3, 5-3=2, then later they may be able to take the leap in understanding that 3 times 6 equals 18 and 18 divide by 3 equals 6.

W1Q1

Knowing math facts is he foundation for math success throughout schooling and into their adult life. It is something used everyday in some capacity. When they do not know their math facts, they struggle with assignments and fall behind when learning new concepts. It takes them a longer time to complete activities. They are more likely to receive lower grades. When they have to count to figure out their addition problems, they are prone to make simple mistakes.

Hi Jamie,

On July 2 I read Math Curse (again), to see through the eyes of a child how math impacts our daily lives. As I read the book, it was wonderful to see how the student made the connection between the abstract concepts being taught in school to her daily experiences. I contend that students are seeking ways to see connection between school and the world outside the building they enter every day. If we are able to guide them to the realization that in school they are learning the vocabulary to help explain what happens beyond the walls, I think we will help the make the teaching/learning process more relevant to them.

W1Q3

I start by giving a personal problem I had (ex. “I wanted to make pancakes and the recipe said that I needed an egg. I also wanted to make scrambled eggs. I needed two eggs to make that. How many eggs did I need to cook my breakfast?”) I like using verbal stories and student participation to show addition situations before I introduce the number sentence and the signs (+,=). I use my student’s names and the things they like (pizza, video games, cookies, Pokeman, sports, American Girl, tv shows, etc.) to tell stories. After I tell the story, I write down the number sentence without going into detail. I leave the equations up. On the first day, I tell 3-4 math stories. I try to do a few more stories throughout the day continuing to write down the equation. The next day, I start with another addition story using student participation. I follow this by using manipulatives instead of students to show the story. I continue writing down the math equation without going into detail. On the third day, we look at all the equations and talk about what each sign means. Many of the kids have figured out what they mean by listening to the stories and watching me write the equation down. To bring in kinesthetic learning, we make arm movements to represent the signs.

Jamie,

I love the strategies that you have shared: stories (auditory learners), hand movement (Kinesthetic learners) manipulative (tactile learners) students’ names (making connections/building relationships), tv shows, cards (visual learners), talking bout math (verbal learners). There are a lot of strategies that we may use to engage students, and the more we engage them, the more meaningful learning becomes.

W2Q4

I like the Plus One Two Bingo game. I would introduce the game when we are in small groups. Each student would get a blank bingo card. I would also have the same card. I would model writing the numbers down so that they could see that I didn’t write them down in numerical order (3, 5, 9 instead of 1, 2, 3). Two helpers would be picked to 1)pick a card and 2)spin the +1, +2 wheel. We would play the game allowing each member of the small group to do both helper parts. Once each student has had the opportunity in small group, I would put this game is math stations. I like this game because you can make small changes that would keep the game from becoming redundant. You can make the bingo card using pictures and borders that correlate with a theme or holiday. You can also change the bingo markers. Instead of using a spinner, you can use a blank dice writing +1 and +2 on the faces. You can extend the numbers as they master new facts (+3, +4, etc.) Besides walking around and observing, I like when students give feedback about their partners and themselves.

I love the fact that you are willing to have students work in groups regardless of their age! Students work in groups all the time when they engage in activities outside our walls such as when they are a part of a sport team. I know it takes preparation and time to teach the expectations, but once the expectations has been communicated, modeled, practiced, and reinforced, I know it works.

I love the idea of using the bingo card and allowing the students to write the numerals On the bingo card in any order that they wish. I also like the idea of changing up the bingo card and using it for the same game. Instead of having numerals On the bingo card we could have the 10 frame representation for that number, we can have super typing dots representing the number, We could use tally marks as a representation, or even an image I have the number representation on hands. I like the idea of being able to teach one game at in small group at the beginning and then being able to change the game or just by changing the we could use tally marks as a representation, or even an image the number representation on hands. I like the idea of being able to teach one game at in small group at the beginning and then being able to change the game up just by changing the way you represent the number.

Kaylene,

Fantastic extension for the bingo cards. What would that look like? Could you draw or create one and share it with me?

W2Q5

I like My Truck is Stuck! written by Kevin Lewis and Daniel Kirk to introduce +1 to my class. In this story, a dump truck gets stuck. As each vehicle passes, they try to tow him out. “Dump truck plus 1 equals 2.” In the book, the number is written in numerical form. When I ask how many cars are stuck, they can say their answer out loud. When I show the page, they can see if their answer is correct. I copied some of the pictures and we retell the story using the magnetic pictures and the white board. When doing this activity, you can change it to add 2 instead of 1. (ex. “The tow truck is stuck. 2 cars drive by and try to help him. How many cars are there now?”)

Five Little Monkeys Jumping on the Bed written by Eileen Christelow is a good book for teaching -1. This is the classic story of monkeys falling off the bed and bumping their heads. One by one they fall after not obeying the doctor’s orders. I use pictures or finger puppets to retell the story focusing on subtracting 1. I use 10 puppets instead of 5 to extend the activity. Two monkey can hit their heads to show -2.

Jamie,

I love connecting math and literature. I believe literature gives the children something to connect and refer to when exploring new concepts. I’ve really enjoyed reading the lessons in this book that focus on literature/math connection.

W3Q2

“Addition is a joining or part-part-whole process” can be explained and reinforced by using manipulatives, pictures, or body movements. “3 bears are playing on the swings. 0 bears are playing on the jungle gym. How many bears are playing?” I love using magnetic pictures!! I use these to create all kinds of stories. “2 turtles are sitting on the rocks. They look all around to see if another turtle would like to sit with them. They see a fish, a snake, and a duck. But they can’t find any more turtles. How many turtles are on the rocks?” Using kinesthetic activities helps keep students moving which helps them stay alert and on task. “Sam and Janet are exercising. Sam does 4 jumping jacks and Sara does 0 jumping jacks. How many jumping jacks did they do?” The class can all pretend that they are Jack and do 4 jumping jacks. They can pretend to be Janet and do 0.

W3Q4

The key to managing math games is to make sure they know the directions and the math concept. Modeling this in whole group instruction AND playing it in teacher led small groups helps students become more familiar with the game. I manage the rotation of the games/math stations using real pictures of the activity and a pocket chart that is displayed where it can be easily seen. I try to do a walk through a couple of times between my small groups to observe. I also use the mic/speakers/head set to listen to what is going on around the room. Thanks to sticky notes and folders color coordinated to their math stations, I am able to keep organized notes. When I see someone who is struggling, I determine if it’s because of the math skill or the difficulty of the game. I then know if the skill needs to be retaught or the game directions need to be reviewed.

I am loving these ideas and am taking notes! How are you using the sticky notes and folder? Is it tracking students’ progress or success with each particular center?

Using the Lightspeed mic and speakers in a great way to monitor them without them knowing you are doing it! I was wondering how that worked in a smaller classroom than my previous. I bet you can learn a lot just listening to them talk an play the games/work with the centers!

Jamie,

I struggle with implementing math centers. I feel pretty confident in creating and organizing centers but when it comes to deciding how often to rotate, should their be records of their work, what if they don’t finish and such, I feel lost. I have trouble determining appropriate expectations in regards to these areas. I would love to see the methods that you mentioned.

Jamie,

I like the book titles you have shared. I love to use books to make those math connections for kids, which gives them a literature base to help them enjoy and remember the mathematical concepts being practiced. I also like the way you gave an example to easily adapt the book My Truck is Stuck.

I also totally agree with you about having kids understand the expectations and procedures of groups before they are allowed to move through them independently. This is not, in my opinion, something that can be taught once. It must be modeled, correctly and incorrectly, so kids know exactly what it is supposed to look like and sound like. I really enjoyed your insights and comments.

Hello all! I am finally back home and in the presence of steady, reliable wifi. Plus, I have a laptop that is easier than using a smartphone to type all of this!

W2-Q1) Reflect on the importance of students’ understanding of the commutative property. In what ways will it support their success?

Students need to understand commutative property as it is a critical skill for them to grasp before moving on to more difficult scenarios. If they don’t fully understand it, a word problem that should take 2 minutes to finish suddenly turns into a 10 minute, frustration and anxiety filled problem. However, by knowing the commutative property, they won’t be “reinventing the wheel” and will have more automaticity in solving problems. This reduces anxiety, leading to less stress overall of math. Additionally, the commutative property is important when they get into higher level math, like algebra and calculus. If they don’t grasp the fundamental concept now, they are going to have a tough time applying skills when they get older. It is definitely a crucial building block in their mathematical learning experience.

W2-Q2) Reflect on the inverse relationship between addition and subtraction. How will this early understanding support students’ success as they continue their mathematical journey?

As a visual learner, addition and subtraction has never been an issue for me. Subtraction is the opposite of addition. Period, no need to go on any further. Yet, for some students, it seems that subtraction becomes this entirely different and unrelated thing that they need to learn how to do. If they can really grasp the concept that + and – are opposites, or inverse, of one another, that will allow them to be able to more fully understand the more advanced mathematical concepts in years to come. For instance, when solving for x in a simple algebraic equation, students must know that when you bring a positive to the other side, it is now a negative, or subtracting. Without a full and complete understanding of the inverse utilization of addition and subtraction, algebra will be a confusing and frustrating process for these students.

Well said, Kristi. I think most teachers think that kids pick up the commutative property and inverse relationships very easily. However, each problem can also be seen, by some kids, as a new and unique situation to solve. It is imperative that we, as teachers, help kids make those connections and build those relationships between the facts. You are so right when you say it is crucial to understanding Algebra and more abstract forms of mathematics. Great comments.

W3-Q1) Adding and subtracting zero is an abstract concept and might be confusing for some students. What strategies might you use to help students build their understanding of adding and subtracting zero?

I am a firm believer in using physical movement to teach tough concepts. The book refers to “acting out” scenarios. I think this will be a go to strategy in my room for a couple of reasons. First of all, for those students that require physical movement in order to really piece together the concept, we are giving them that opportunity. Secondly, those students that rely on visual representations have a very large and real scene in front of them. For some students, they have to not only write things down, but physically take part in completing the concept. I am one of those people and if you watch me solve a math problem, I am moving my hands, mouth and eyes. I cannot sit still while doing math, but it helps me visualize the processes being used in order to solve the problem. I feel like students should be encouraged to use their strengths in order to grasp these skills. If that means we are having students standing on one leg and wearing pink boas to pretend to be flamingos and then add 0 to them to demonstrate the concept, we will.

W3-Q2) On page 52, there are four “big ideas” presented that will help with our exploration of adding and subtracting zero. Select one of the big ideas and suggest a hand-on or kinesthetic activity that you could use to help support that idea.

“The order of addends does no change the sum (the commutative property).”

With this big idea, I think it would be useful to act it out with the students.

For example, there are 6 birds in a tree, during a storm, 0 more birds join them in the tree.

I would have 6 kids come to the front to be our birds. We have the “storm” (gently and quiet blowing wind) and then add 0 more students. How many birds (students) do we now have total?

If we switch it now to 0 birds in a tree and during the storm, 6 more birds flew into the tree to join them, it is the opposite. We start with 0 students at the front, have the storm, and then add our 6 students. How many birds are there in total?

This would be an activity that students could even recreate in a center. Instead of using students, they could use figurines or manipulatives, resulting in a transfer from teacher-led to student-led.

W3-Q1 – I always thought that adding zero would be one of the easiest concepts for a student to grasp. I thought it was interesting that understanding +1 and +2 facts prior to introducing adding + O makes understanding +0 easier. Kids learn through movement, therefore acting out +0 would be just as important as acting out +1 and +2. Creating scenarios for the kids to act out and also providing opportunities to create their own scenarios allows them to do their own mathematical talking and writing.

W3-Q2 – Referring back to the “big ideas” as our children are exposed to +0 will hopefully help them understand that +0 follows the same rules within the “big ideas” as +1 and +2. One idea would be to have the children take objects and place them inside a hula hoop and then have them “put zero” in the hula hoop. They would be able to see and participate in the actual change and lack of change when adding zero.

Jodi,

I love having my kids acting out the action. It really helps them make better connections because it involves movement.

Jodi,

I totally agree. I always imagined that adding and subtracting 0 would be so easy to teach and for kids to understand, yet it seems to be one of the more difficult concepts for them to understand. We do a lot of acting out when learning this concept.

W3-Q1: Adding and subtracting zero is really abstract for some students. I like to get my kids moving and acting out the process, drawing pictures to represent the action, using base-ten blocks or other materials to show the action etc. Once students have more of a visual or they have had a chance to experience the action they tend to make better connections.

W3-Q2: The order of the addends does not change the sum (the commutative property) I would call some students up to the front of the class and have them help me model 4+0=4 and then 0+4=4 then I would have students work together with a partner to draw comic strips showing the action of different algorithms given.

Kelly,

I love the idea of comic strips. What about students who do not like to draw? What options could be provided?

W1 Q1- Mastery of basic math facts is important because it is used throughout daily life into adulthood. If students don’t know them it will frustrate them and slow them down later. Once students get to third grade, if they don’t know those basic facts then it will affect their mastery of multiplication as well.

Melissa,

When you meet students who are struggling with basic math facts, what strategies do you use to help them?

W1-Q2 – I’ve seen anxiety related to memoring facets in both 2nd and 3rd grade. Many times students seem to have a block when they are trying to solve problems without the basic math facts mastered. If they had to be in a speed competition it would be very frustrating to them and would cause them to shut down.

Melissa,

How do you help students to reduce their anxiety?

W2-Q1 – The commutative property seems to be critical to students for mastering those basic math facts. Once students learn these properties then the memorizing of their facts becomes easier and will build confidence as well. In second grade and third grade we teach the importance of those relationships which becomes automatic for them.

W2-Q4 – I like the Hop the Line game because it gets the students moving and it is able to be used for addition and subtraction. I would introduce it as a whole group to the class and allow all of them to play before putting into a rotation. I like that one student is acting it out while another is telling them what to do.

W3-Q1 – Adding and subtracting zero is abstract for many students. Having students use manipulatives such as base 10 blocks or counters helps them to see it in action. Also having them act it out or draw pictures can be very helpful. This allows them to make better connections once they have had the opportunity to do it in these ways.

W3-Q4 – Managing math games in the classroom starts with good classroom management and clear expectations. By going over each rotation with the whole class the students know what they are supposed to be doing when they get there. I like having color coded bins for each station so that they know which one belongs in each and it helps to keep them organized. If I notice someone is struggling with a concept then I will make sure to address it in small group when they are with me during their rotations.

I agree that successful math rotations depend on good classroom management and clear expectations. We begin our year with teaching students the expectations of math rotations and how to switch while exploring with manipulatives. Once games are introduced to the class in small group, we slowly add the, in to the rotations.

W1-Q3

I introduce math facts to my students using real life situations, or using math found in literature. We can talk about how many boys, how many girls, how many friends all together? What if we got one more girl in our class? What if we got one more boy? How many cookies do you have? If I give you one more, how many would you have? Math is important, and it is all around us. By noticing and talking about the math problems we see around us, we can help our students to think like mathematicians from a young age. One more and one less can be a hard concept when just talking about numbers, but if we are talking about real cookies, we know right away if someone has one more and we have one less.

Emily, I agree that using the students and real life examples helps students understand math concepts. I find that we often use these real life experiences first and then we follow up teaching that concept using numbers. In a way, it’s building their mathematical schema.

Emily,

Any lesson using meaningful experiences yields success! One my students’ favorite activity involved an aquarium with fish (goldfish) which they got to eat for subtraction.

W1-Q5

Anytime food is involved, we can use math. How many would you have if someone gave you one more chicken nugget? I had 5 chicken nuggets and I ate some, now there are 2. How many chicken nuggets did I eat? I had 2 cookies and my brother has 3 cookies, how many do we have all together?

We can use math to figure out how many supplies we will need at our tables. We have 4 friends at the table, and 3 pencils, how many more pencils do we need? Or, we have 4 friends and 5 pencils, how many pencils should we put back in the pencil box?

Emily, I like the practical approach to math as well. Whenever we are able to use food it tends to trigger some form of connection to the real world adding value and relevance to the subject. As facilitators in the classroom, we need to help create opportunities for students to problem solve, such as figuring out how many more items they would need at their tables…another great idea to increase meaning and relevance in the classroom using basic facts.

I love your focus on day to day items that the students are so familiar with. By using items that they are familiar with, mathematical learning is so much more meaningful. The kids always love to do addition and subtraction problems using snacks as the manipulative.

W2Q1 – Once students are able to understand the commutative property of addition, they instantly know twice as many math facts. When they have mastered adding one more, and adding two more, and understand that the order of the addends does not matter, they can also add 1+___ and 2+___ facts with ease. We can teach our students to begin with the larger quantity and then add 1 or add 2.

W2Q4 I liked the game, Hop the Line for practicing +1 and +2. It could be demonstrated first as a whole group activity, one student chooses a number card, a second student hops along the number line to that numeral, and the first student spins the spinner to determine +1 or +2, then the second student hops again accordingly and shouts out the answer for everyone to record. Then 2 more students can take a turn. It can be repeated several times, and all students can be recording the number sentence created. Then this game can be moved to math tubs or stations for group or partner work.

Emily

I also loved the hop the line game. I have used this game for more and less by putting painters tape on the cabinets to build a number line, but have not actually put the number line on the floor. I really want to find a space to build a floor number line so that the students can connect the movement to 1 more and 1 less.

Kaylene and Melissa,

Students recording the equations is what I love about Hop the Line. Disruptions always arise when students are just spectators.

W1 Q1- Mastering basic math facts with automaticity sets the foundation for a lifelong learning in math. In kindergarten, we are working on building that foundation. Every once in a while I get a student who knows addition or subtraction, in actuality they have memorize the math facts, but do not actually understand the operation that is happening. We then start from ground zero using manipulatives and allowing the student to understand the process that they are actually doing.

W1 q3- In kindergarten, we teach a lot using children’s literature. I love to start as many lessons as possible with the story. I love to read to the students and have them discuss what is happening and eventually make a prediction based on the pattern and the back. This allows the students to Easley talk about math concepts. Once we move onto a guided activity or independent work. You get a handful of students who make the connection to the story and then “math talk” appears.

Kaylene,

I have found parents get so excited telling me their children can count or multiply, when all the child has done is memorized facts. They do not understand one-to-one correspondence, cardinality, magnitude and forget about subitizing! Rote memorization is a start, but that cannot be the end.

The use of manipulatives is huge. I love your use of the number line, especially when you “cannot keep the numbers on” and the students have to keep on helping you 🙂 All of that creativity is helping to make math memorable and that is where we want them to start.

Kaylene,

I agree that understanding needs to come first before memorizing facts. When we work on math problems using addition and subtraction, I like to model using magnetic objects while they are drawing pictures. I can observe who is understanding addition as joining and subtraction as separation.

Emily and Kaylene,

I enjoy reading your posts as you respond to Math Facts. Keep up the great work.

W2q4

I really liked all of the game ideas in the back of the chapter. We spent a lot of time on one more two more one last to last and kindergarten and sometimes it feels like it’s difficult to continue coming up with game ideas. What I like about the game ideas and this park is that I can find a bunch of materials that we use during numeracy stations why recognition or super ties in cards or even 10 frame.cards. What I like is that I can use those materials and then just add a spinner To decide if we are figuring out one more, two more, one less, Or two less with the help of manipulatives. I also really like that whenever they do these games, they usually always connect at where an addition or subtraction number sentence. I find that at the beginning of the school year that would be difficult, but when we revisit these concepts later on in the school year it’s a very important thing to do you know that we often overlook because we are wondering why we are still teaching one more and one less at the end of the year. Now I know that the key piece is attaching those addition and subtraction equations.

Kaylene,

I contend that students need the experience before we start labeling everything. In kindergarten, it is important for them to build conceptual understanding through interactions with manipulatives in numerous ways before we start introducing them to number sentences. They will have their entire lives writing and creating number sentences, but in kinder we get one year to make them fall in love with mathematics and the many possibilities that this new language offers.

Effective learning requires us to revisit and extend understanding anyway. We certainly have time to introduce concepts and then extend and challenge their thinking later.

Kaylene,

I understand that at the beginning of the school year, it is harder to begin games. However, we can make the games simple. For example, tower building for counting or number bingo for recognition. What I usually do in the beginning is have activities in math centers that involve manipulatives.

W2q5

Fish Eyes- This book addresses counting that as the book continues the original finish at himself to the page. This is a great book that are used to work at a and then add one more because of the original fish to see the total amount of fish that are on the page. I also use the same book later on during shapes to have the students just spot different two dimensional shapes that they see on different fish.

Pete The Cat and His Four Groovy Buttons- All of the kids seem to love Pete the cat. This is a great book that works on one less starting with the number four. This book actually ties the subtraction sentence to each time Pete loses a button.

W3q1

Adding and subtracting zero is definitely a difficult concept for students to understand. I find that the best way to teach students desk concert is to actually have the students act out the problems. When first introducing the concept are used boys and girls to act out a story problem. For example Mrs Foster has three girls setting at her back table and Mrs Foster has zero boys sitting at her back table. How many students are sitting at the back table. Once the students have had an opportunity to physically get up Dan and front of their students and experience adding zero more, we then do you guided practice where students have goldfish or cereal and we practice adding zero and subtracting zero that way. My favorite problem and you have three goldfish and you Give zero goldfish to Miss Foster’s how many goldfish do you have left? After we have acted out different problem than we connect the problems were at an addition or subtraction sentence.

Kaylene,

I think it helps to introduce adding or subtracting zero when the students have mastered the concept of +1/+2 and -1/-2. I like how you use kinesthetic activities. I use them too. For example, I ask the boys to line up and then I say zero girls line up. Illustrations also help and the plates with compartments to use with word problems.

W3q2

The big idea that we focus on the most in kindergarten is addition as part-part-whole. We focus on this concept early on with composing numbers and the students become familiar with the idea of joining groups. One activity that we use are part-part- whole plates. We use the 3 section plastic plates to work on addition problems.

This would work great with adding 0. As an addition problem, is read aloud, students place manipulatives in the 2 small sections of the plate. Then students mo e the manipulative side from the 2 parts to the large portion which represents the whole.

W1-Q2) What have you observed about anxiety related to memorizing math facts? Are there types of math facts practice activities that increase anxiety or decrease anxiety?

I have observed anxiety when understanding is not connected to the task. It doesn’t make sense to memorize something that is not understood. Understanding must be developed first with memorizing facts coming later. The types of math facts practice that increases anxiety is when a student is asked to come to the front and explain a concept. Timed activities (speed games) play a significant role because stress blocks the flow of thinking. To decrease anxiety, I allow partner discussions during whole group instruction, working with a partner in math centers, rotations, small group instruction and kinesthetic activities.

W1-Q5) What real-world experiences might create an effective context for addition or subtraction problems?

I have used counting red-hawk bucks to buy from my treasure box. The students realize that as they spend their bucks they stay with less. I also created a shop bringing a cash register. A student was the clerk selling fake food, students lined up to buy 2 items using only pennies to practice addition.

Tammy,

I offer that taking time to develop understanding is key to boosting students’ confidence in math (or in anything they are called upon to complete). I think we may all recall when we were called or sent to the board and how anxious we felt even if we knew the answer. The increase pressure of knowing everyone is staring makes the individual vulnerable and the increase fear reduces long term memory. The very thing the student was called to the board to complete may escape his/her memory due to the act of being in front of the board.

I agree that allowing students time to think about their responses and to talk with a peer will help to reduce anxiety level and may yield better results.

Emily, Kaylene, Tammy,

Melissa talked about liking the Hop the Line activity. How would you use this activity in the kinder classroom?

Mrs. Johnson,

We have used the Hop the Line activity by having a giant number line placed on the floor. The blank number line has a magnetic piece where the students place the corresponding number cards in order. I use popsicle sticks to call out two partners. One student rolls a die and hops to the number rolled and the next one uses the spinner +1/+2 or -1/-2. The first student then jumps forward or backward. The second calls out the sum and records the equation. The rest of the students also record the equations.

Tammy,

Thank you for sharing. I look forward to seeing such awesome activity taking place again this coming school year. Please remember to invite me.

Sherry J

I think it could be used with all math facts. I would introduce the game in small group then move it into a station. I can change the math facts out. Plus 1 and plus 2 together. Minus 1 minus 2 together. I also think it would be great with 0.

W2-Q1) Reflect on the importance of students’ understanding of the commutative property. In what ways will it support their success?

The understanding of the commutative property will help students to build automaticity because they can add by counting on from the larger addend and then add 1 or 2. When Kindergarten students understand this concept it will help in the future with multiplication which also uses the commutative property. Students will understand that it works only with addition and multiplication because we are increasing numbers.

Commutative property and place value are also related. Place value and commutative property are important to remember when understanding and solving addition and multiplication equations. The order of the numbers in the equation does not matter, because the sum or product is the same.

W2-Q4) After reading pages 41- 50, select an activity/game that you could adopt or adapt to

help students build automaticity. Explain how you would introduce it to the class and monitor students’ progress towards mastery.

To help students with automaticity, I would use fact checks having the students use the same worksheet numerous times to build fluency. Students can record their own progress and get motivated by setting goals and beating their own record. They track their own progress using a recording sheet. Students record the date and the number of facts answered correctly.

W3-Q1) Adding and subtracting zero is an abstract concept and might be confusing for some students. What strategies might you use to help students build their understanding of adding and subtracting zero?

I will make sure that the students have grasped the concept of +1/+2 facts and have developed a solid understanding of addition and subtraction before I can introduce 0 facts. This concept would be more easily understood. Using addition and subtraction word problems as well as plates with compartments, students are able to see that nothing is being joined or separated from a set so the quantity always remains the same. Students can act out a word problem, illustrate it and come up with problems of their own. For example: There are 3 baby birds in a nest. No more birds have hatched. How many baby birds are in the nest? Other activities may include: Hop zero times, blow 0 kisses.

W3-Q4 How will you manage math games in your classroom? How will you keep track of students who are experiencing challenges?

When math games are involved, the instructions, rules and behavior expectations need to be modeled, explained and practiced before they go to centers. They will be modeled using the document camera or we can also practice playing for a short period of time. Then the students can go to their tables and play with a partner while I walk around observing and writing down in my monitoring notebook who Is experiencing challenges, so I can help them during small group instruction. Behavior expectations for games is very important too since it can get loud when playing games. I will model an acceptable voice level, time limit, taking turns, where to place their recording sheet, cleaning up and the consequence for not following the expectations. Taking away the privilege for that day if rules are not followed is something I would choose.

Kaylene,

I will certainly incorporate those two books you mentioned. I have used for addition “Ten Apples up on Top”. Students then use books on their heads and compete who can carry the most without dropping them. For subtraction, I have used “Five Speckled Frogs” as a song too. “Five little Ducks”, “Ten in the Bed”. The book “Ten Little Apples” by Natalia Vasquez is great because it has a dial that when turned the students can see the apples falling down. One of my favorite books that show addition and then subtraction at the end is “Monster Math” by Anne Miranda, Polly Powell. A monster is celebrating his birthday. His first guest arrives. and now there are two, three, four! The guests keep arriving until Monster Mom decides they should depart until just one remains. I highly recommend “Animals on Board” by Stuart Murphy. This book actually has the math equations written on the pages. The pictures are hilarious and the words rhyme.

Kaylene,

I love those compartment plates too. They are ideal for addition. I use them more than than the part-part whole sheets. I also model using a giant hula hoop. I place a cord in the middle and staple a plus sign. We come up with a word problem, I call on a student to represent the addition while the rest record the equations. To support the big idea” Addition is a Joining or Part-Part Whole Process” , we use “Add/ Zero Cover Up” which is a partner game using a game board with numbers and 2 sided counters. Students take turns rolling a die, add zero to that number and cover one of the answers( numbers) on the game board. If it”s already covered the student loses a turn. They play until all spaces are covered. The one with the most spaces covered wins the game.