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In Kingergarten, we use math manipulatives  quite often, especially when learning addition and subtraction. A couple weeks ago, I had been teaching how use different manipulatives one at a time, which worked for most students, but there were a few students who were still not understanding the concept of subtraction. I realized that rather than telling them which tool to use, that I should give these students the option to use what works best for them. During a small group lesson, I was encouraging students to use their fingers to solve subtraction equations, but one student was really struggling with this. I asked him if he would like to use something else instead, and he chose unifix cubes. After working with him on this for several minutes, I could see that this tool made much more sense to him than using his fingers when it came to the concept of “taking away”. It is so important to listen to our students because they will often tell us exactly what they need.
Jess W

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I have three different rulers in my classroom. I have the Math Expressions rulers, 12- inch wooden rulers, and 6-in clear plastic rulers (0 is not lined up with the end of the ruler). When I began working on measurement with my class, I was passing out rulers and assigning students with the ones I thought would work best for them. After a few lessons, I started leaving the rulers out on the table and allowing students to pick their own- something I should have done all along. Something so simple excited my students! The students were successful using each kind of rulers. The students even often chose different kinds of rulers each day rather than always taking the same one.

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When teaching students about Math Mountains in our Math Expressions program, I tend to focus on what learning opportunities are there in black and white. I believe it would have been a great idea to let students construct their own knowledge by modeling them with math manipulatives.

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I am pretty set in using manipulatives in math class for place value. However, there are innumerable options for developing the understanding of a base 10 number system. I always fall back on the tried and true foam blocks. However, I think I should be more flexible with items like unifix cubes, pennies and dimes that sort of tool. It is important for my kids to see the repetitiveness of using the same tool for the same type of problem, but they struggle to generalize skills when I get stuck on the same material. This is an important lesson for me when I present lessons. I need to be flexible and let my students guide this at times.
Robin

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We recently completed a few lessons investigating decimal fractions. A few lessons relate tenths and hundredths to pennies and dimes. Useful lessons and we used tools such as math boards and secret code cards during the lessons. Then I was thinking why not use pennies and dimes (plastic ones) instead of just discussing them or using illustrations. I think it will be beneficial to the students. I'll use these to manipulate and work with tenths and hundredths in the future.
John

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I feel like sometimes I hinder students by insisting that they use manipulatives, when they might not need them.  On page 80, in chapter 5, the author writes that sometimes students need to be reminded that mental math is enough.  My high math students don't always need to use clocks and timelines to figure out elapsed time.  Therefore, am I making something more difficult for them when I insist they show how they got their answer, when they ALWAYS get their answer correct using mental math??  Food for thought.  I think it depends on the skill, however.  We have had many discussions about students who have memorized facts but don't really understand the "how" for finding the answers.  So, again, I think it depends on the skill involved and, again, food for thought

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I feel like sometimes I hinder students by insisting that they use manipulatives, when they might not need them.  On page 80, in chapter 5, the author writes that sometimes students need to be reminded that mental math is enough.  My high math students don't always need to use clocks and timelines to figure out elapsed time.  Therefore, am I making something more difficult for them when I insist they show how they got their answer, when they ALWAYS get their answer correct using mental math??  Food for thought.  I think it depends on the skill, however.  We have had many discussions about students who have memorized facts but don't really understand the "how" for finding the answers.  So, again, I think it depends on the skill involved and, again, food for thought

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Else

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Students worked in partners to solved 2-3 digit word problems. They were prompted to grab the tools they would need to solve the problem. Some students grabbed counters and quickly discovered that wasn't the most efficient way. They traded those in for place value models. Other students who used place value models discovered that they needed more tens and hundreds to complete the problem. It was fun to see when the light bulb went on and they were able to represent their work with their models and their models were able to help them solve the problem.

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It is very common for me to be found pulling out unifix cubes when it is time to teach multiplication.  Students can easily set up arrays.  Clean up is generally quick as they simply snap them back together in ten strips when we are done.  Fast forward to teaching finding area of a rectangle.  Here I go again.  You generally find me pulling out 1 inch square tiles.  It makes sense.  Right?  Students are able to create a rectangle with the tiles, and they are generally flat giving the impression of a 2 dimensional figure as opposed to a 3 dimensional shape.  Sounds good so far.  However, I find that when I'm teaching area of a rectangle, some students are quick to memorize the formula of length times width.  However, they don't fully understand what they are calculating - the 2 dimensional surface area of the rectangle inside the border or edges.  I wonder if I were more open to providing a variety of tools when teaching multiplication, if they would grasp the concept of surface area of two dimensional figures at a faster rate.  What if they had used the one inch tiles to learn multiplication/building arrays?  Maybe the connection between arrays and surface area would have been more clear.  Many students are usually able to see the progression of their "unifix cube arrays" for basic multiplication to their 1 inch tile arrays for area.  However, for those that simply don't see the connection, I wonder if it's because the unifix cubes are three dimensional and the tiles are 2 dimensional.  Maybe giving more options could bolster some of these connections.  It is so difficult to see inside the thinking of a young learner and figure out why they aren't understanding.  Maybe we need to rely on them to help us a little bit more.
-Wendy

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During our unit on fractions, students need to be able to compare fractions. The book provides fraction strips for the students to cut out and this is the tool I presented first. Some students had difficulty lining the strips up correctly (esp if they were wrinkled and bent) so moving forward my students were given the option of using a whole sheet with fraction strips on it. I also introduced the fraction stax as a tool that students could use. Many found these helpful because they could physically pick up the unit fractions to build the parts they needed and then see them stacked up.

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I agree with Kristi’s experiences with using math tools for elapsed time. When first faced with elapsed time problems, my students want to use a Judy clock (or invisible mental clocks!) to find elapsed, starting, or ending times. As you could imagine, this leads to a lot of errors, especially when several hours are involved and the hour hand doesn’t move accordingly. I teach the time line strategy from Math Expressions, and they also must use this strategy on district assessments. This does seem to be the most successful method for many kids, since it makes an abstract concept appear more linear. However, I wonder now if the kids really do understand why this tool works. Do the kids understand the connection between a circular clock and a linear time line? It could just serve as another shortcut for them. Using a Judy clock forces the kids to physically act out hours and minutes passing by, so perhaps I shouldn’t shy away from this tool for elapsed time as much as I have in the past. By pushing them in the direction that I find most helpful, I could be taking away the opportunity for students to “get it”, not just get it right. -Carolyn

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An area that many second graders struggle with are triads. Even though we learn about place value and comparing numbers, students have a difficult time determining whether a specific number is closer to one number or another. We practiced this concept in class and during Calendar Math,  but students still struggled when it came time for AimsWeb testing or progress monitoring. A tool that we started to incorporate is a hundreds chart. This helps students to visualize the three numbers they are comparing.
-Ashley S.

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Gary and I team teach in math and I have learned so much from him.  He "gets" that students need more concrete examples.  I rarely think to pull our manipulatives without having planned for them.  Gary creates manipulatives where there are none, which I think is so cool!  We were talking about metric liquid conversions and Gary started going around the room scarfing up water bottles to show the difference between liters and milliliters.  The majority of the class "got it" instantly-ish.  It was a pretty awesome "ah-ha" moment!
~Beth

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As an adult, I do have to admit that there are times when I have chosen the tool for the student to use for a specific skill.  I am trying to move away from that and allow the students to select what works best for them
When we were working on an elapsed time problem, I gave the students clocks and asked the to use them to find the time that had passed. While many were able to do this and were successful, there were still a few that struggled with it. I allowed them to work on the problem and found that they were able to do the problem with a number line. Since this was a "tool" they were comfortable with, I allowed them to continue without the clocks.
- Julia

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One of my third grade intervention groups was still really having a hard time with regrouping with subtraction. I tried lots of strategies, even going back and using base ten blocks like the first time it is introduced in the primary grades. My one group in particular was still having issues and didn't quite understand enough to perform with consistency. It wasn't until I overheard a student tell me that she loves money and can  understand anything about money because she loves it so much that I had an obvious ah ha moment that I could link subtraction to money. The next class, I opened up my 'store' where students had play money and had to pay in exact change, and if they didn't have exact change they had to go to the bank to make an exchange (one dime for ten pennies and eventually one dollar for ten dimes). We didn't talk at all about subtraction, and just spent the whole class doing this. Next class, we made the connection with the traditional algorithm, and things are slowly improving!
-Dustin

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For the AIMSweb testing, the questions based on triads seems to have my students, along with many other second students, struggling with  the concept. I never thought of allowing math tools, such as a 100 chart, to help students with the questions, especially for a intervention test. I have since implemented more place value questions in my Calendar Math without asking questions specifically about triads (teaching to the test). The 100 chart has helped with a few of the problems to help visualize the numbers, but students still struggle with numbers past 100, hence the additional teaching of place value.
-Ashley C.

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Currently, we are working on using benchmark fractions to compare fractions, and I'm using dry erasable number lines.  We currently took a little quiz to check for understanding, and I had them use the number lines although I would like to find another manipulative to use possibly to help the struggling learners.  Some students already know how to compare fractions without using the dry erase number line, so they are using that strategy.  Without showing them how to find a common denominator has been hard for the students that aren't grasping the concept of the benchmarks and how they relate to the fractions and mixed numbers.  I feel that I have made students use the manipulative that maybe I could have left them to use their own strategy and were looking at me with this, "What is she trying to get me to do" look, but yet struggling with how to get the others to recognize the relationship of the number lines to the fractions. and using the manipulative I had for them.    Jaci

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When I introduce these skills to my students, we use drawings with their math boards.  I would like to add cubes to help those who need a more hands on learning experience.  I would also like to introduce using a number line.  My students all have a name tag on their desk with a number line.  I would like to teach them, earlier in the year, how to use the number line to find the answer for the equations.

-Linda

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Recently we worked on a math unit involving elapsed time. During this unit, students take a quiz where they need to use a timeline to solve a word problem. Students have learned numerous ways, by the time of the quiz, to solve an elapsed time word problem. I have seen in years passed that timelines are hard for students. We teach students that they can make t-charts, use clocks or timelines based on what works best for them. They can also use more than one method to check their work. However, this quiz is hard for them based on the fact that they are given only number lines to use. I would like to change this for next year. Choice of manipulatives helps students to create their own learning and find what works best for their understanding. MP6 would back up the fact that students need to be able to show their thinking and logic based on methods that they can explain logically. To do this, students should use manipulatives that help them reach their goal.
Sophie

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We were recently practicing multi-step word problems, and this problem happened to have about 5 steps. We read the problem a few times, underlined what we thought we needed to know, and discussed what was happening. I did this first with my high group and it was a struggle! The students were not understanding what was happening in the problem. We tried drawing pictures and talking it through, but it still wasn't clicking. When I got to my other groups, we practiced the same problem. This time I used counters to represent the problem. The students and I modeled what was happening with the counters. This allowed them to see what operations were being used and how the equation should be written. I definitely need to make counters and such available on quizzes and tests!

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This year we introduced a new concept in number sense, triads. Triads are problems where the students have have 3 numbers and they have to decide if the middle number is closer to the first number, closer to the last number or right in the middle of the two numbers. As the students completed this assessment it became clear that many of the students in second grade were still struggling with number sense, especially in regards to 2 digit and 3 digit numbers. As the year has progressed we have started adding in more strategies to help the students solve these problems including practicing them during calendar math, using secret code cards and using a hundreds chart. Although this has started to help, the students still need more tools to help them understand number sense, especially in regards to triads.
-Julie

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Finally got into Padlet after multiple attempts on multiple devices. :0) I was often very specific about which tools worked best for each type of problem, but would add "if you find a tool that works better for you, you should use it." In the future, I'm excited about using the backpack idea and, instead of adding the option of picking a different tool as an afterthought, starting the discussion by saying, "here is a tool I have found to be helpful in solving this type of problem. What tools have you found to be helpful? Is it something we already have in our backpacks, or is it something we should add?" This will help my students practice determining the tool that works best for THEM, a skill we adults use on a daily basis!
-Carrie

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I have allowed my students to use the math tool that works for them to figure out elapsed time. However, when some of them had trouble with the tool they chose (meaning they didn't come up with the correct answer with the tool they chose), I would "make" them go back to using an open timeline to figure out the answer.  Ideally, I would spend time looking at each individuals strategy and evaluate what their error was with their strategy.  Then help them fix the error in their strategy instead of having them use the open timeline strategy to answer the elapsed time question. I think I had them use the open timeline because it has a "higher success rate" versus other "math tools" that some students chose to use.  Also, it was quicker to find their errors.   This being my first year of teaching third grade, I look forward to teaching elapsed time again next year with some new ideas and math tools!
-Kristi

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I have been reviewing the topic of measurement, conversions between units, solving perimeter, area and volume problems, etc. Recently the children were converting both customary and metric units of linear measurement. Students were simply making calculations on paper, solving problems that were displayed on the Smart Board. This would have been a good opportunity to distribute meter and yard sticks for the children to explore measuring objects in the classroom. Once students made a few measurements they could have then made conversions.
-Matt

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Every year when I teach the concept of regrouping in addition or ungrouping in subtraction, I always have the students use their math boards to make proof drawings.  Until now, after reflecting on this reading, I have never thought to just give them place value blocks and actually allow them to manipulate the blocks instead of making a drawing.  I think this would be very beneficial for the students struggling to visualize the concept of adding tens or ungrouping hundreds, etc.  

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Every year when I teach the concept of regrouping in addition or ungrouping in subtraction, I always have the students use their math boards to make proof drawings.  Until now, after reflecting on this reading, I have never thought to just give them place value blocks and actually allow them to manipulate the blocks instead of making a drawing.  I think this would be very beneficial for the students struggling to visualize the concept of adding tens or ungrouping hundreds, etc.  

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-Diane