Are You Teaching Branching? Make Sure You Do This First…
Several years ago I worked at a charter school the first year it opened. The school implemented Singapore math, so that was my first year to test the waters of Singapore math. Our trainer instructed the 3rd grade teachers to go ahead and teach branching even though it was a skill the students should have learned in second grade. To teach children the procedure of branching, it took about four weeks total, and then not all of the students perfected the ‘procedure’ of addition and subtraction branching. The students had more success learning addition than subtraction branching. With the mandates of testing, we weren’t able to solely use Singapore math, but I had to supplement with other materials. Then as you are all familiar with, testing approached and likewise the pressure along with it. Then we didn’t have ‘time’ to teach number sense SO deeply since other skills are tested. Unsurprisingly, the teaching of Singapore Math somewhat fell apart midyear. Please don’t take this wrong I LOVE Singapore math because it works, but the conditions of testing hindered us from teaching it wholly.
Fast forward to four years later. After teaching small groups today, I have reflected on the year that I taught branching and its effectiveness. Yesterday I pulled small groups of average math students to teach them regrouping for the second day in a row, I had them build double digit numbers with base ten blocks blocks. I repeated this process today with the same group of students. After that I started notating their thinking with branching representation on a small white board. Students intently watched and helped me notate the thinking they had done with the blocks in (abstract) numbers . They began to understand grouping with tens and how to decompose numbers to build more tens or hundreds. Then I told them that they couldn’t use paper or blocks, but could only look at the addends I was about to write on the board. I asked them to whisper the answer in my ear so that others could still think. I was amazed! Half of them could answer the question correctly doing mental math. The other half were only 1 away from the correct answer. I was so proud.
I shared the above to really say the kids taught me something in just two days because of their adept ability to add mentally. Teaching branching worked so much better four years later–all I had to do was provide an experience with branching directly after building with base ten blocks. Why didn’t I start out with the concrete blocks first before I threw abstract numbers at them…duh me! Branching made so much more sense to them after building a concrete foundation. Reflection is priceless!
Have We Damaged Our Children’s Ability to Reason by Teaching Procedures?
I have been working with many small groups of average leveled students to help build their number sense since there will be so many gaps in common core understanding with students moving up to 4th and 5th grades. Up until earlier this year I confidently thought that students in second grade needed to learn the traditional regrouping algorithms for addition and subtraction. After some common core training, I humbly realized that students aren’t necessarily expected to know how to complete this traditional procedure, but rather be able to make sense of a problem by decomposing and composing numbers using strategies that are comfortable for them. I said all of that to say that because of this thinking, I gave students in my small groups base ten blocks and NO paper. I gave a double digit addition problem that would give students the opportunity to regroup 73 + 48 (mind you they should have been able to handle larger numbers). I thought I would start with a really easy problem. Well, I was wrong! The students struggled to get an accurate answer with blocks to solve this problem. Of the four students I had in this particular small group, only one student was able to find the correct answer. Well, I changed my mind on the no paper and gave them a sticky note. I instructed them to write their answer on the sticky note just so I could do a quick assessment of who could accomplish finding the answer with the blocks without blurting it out. To my chagrin two of the students were trying to solve the problem on their sticky note with the procedural algorithm. I promptly reminded them while replacing their sticky notes that we were solving the problem with the blocks and not paper. When all students had finished thinking, I listed all of the students’ answers on the board and asked them who was correct. After having students count and recount their blocks, they finally came to a consensus of the correct answer …121. They struggled with counting past 100 especially by 10’s… 110, 130, 140… and they would correct one another to say 110, 120, 130, 140. Surprisingly enough to me, the students felt more comfortable lining up numbers in an algorithmic procedure with no understanding to obtain an answer than they did counting out blocks past 100 to obtain the correct answer.
Below is a picture of the students counting out their blocks. I had them place their addends onto small sheets of square paper to help keep them organized since they were getting their extra blocks confused with the ones they were counting. I wanted to use small paper plates to help them organize their blocks, but I didn’t have any. I happened to have some origami paper lying around, so I just used that instead.
Base Ten Block Addends Grouped on Origami Paper
Last Day to Get Your Goodies at TPT
Make sure you stop by the Teachers Pay Teachers today to pick up all of the items you’ve been swooning over while the sale lasts. The sale will be on until midnight tonight, May 8th, CST. Many items including all of mine are on sale for 28% off. HURRY!
Math Cardsorts…Free Addition With No Regrouping Sort
I just finished a product that I posted on Teachers Pay Teachers. If you buy it before tomorrow, you will still get a chance to get it at the sale price. This product contains 8 sorts with addition and subtraction both with regrouping and without regrouping. Some of the sorts contain matching word problems and number disc picture cards. Others contain matching equations and number disc picture cards. In each sort, there is an extra card so that much discussion among student pairs can revolve around the common errors that confuse students with regrouping. These sorts were designed for use after adding and subtracting with number discs. Number discs are one of my favorite ways to teach addition and subtraction with regrouping. They are right up there with base ten blocks. I can’t decide which tool I like better. Number discs are a bit more abstract for students than the base ten blocks. Base ten blocks are the size of actual ones, tens, or hundreds which make them more concrete. Number discs are all the same size–but much easier to draw. For a FREE sort, you can click on the sort below to download the addition sort with no regrouping. The link will take you to the TPT site. Just download the preview for the free sort. I hope you enjoy it.
Woo Hoo! Teacher Appreciation Sale on TPT!
Here is your chance to stock up on many fabulous goodies for your students on TPT before summer is here. I will be offering all of the items in my store for 28% off on May 6th- 8th. Also, many other sellers will be offering their items for 28% off as well. Just grab the promo code below as you check out.
Do Your Kids Need a Dawning Division Moment?
After sitting in a week’s worth of meetings about Common Core instruction and best math practices, I feel as if my way of instructing students is gradually morphing into a new beast, which I am not sure I fully can describe yet. In the midst of all of this, I went back to the fifth graders I have been working with on the day after these meetings. I decided to try out some of the teacher practices I had been learning. Sidenote: This set of fifth graders isn’t yet proficient at long division or ladder division for that matter. I posed a long division problem to the students, but instead of giving it to them in “naked numbers” alone, I gave it to them in word problem form as well. In reflecting, I should have started out with the word problem alone and let the students figure out that they were supposed to divide. After watching the students struggle with solving using ladder division or the traditional long division algorithm, I encouraged the students make equal groups to see if that helped them. One student in particular that I worked closely with (whose work is pictured below) managed to find a correct answer using equal grouping, however he could not find his mistake when he used ladder division. Another student…one I might add who typically doesn’t have much perseverance to solve problems…related completely to the contextual situation and persevered through to solve the problem to the end. To get to my MAJOR DIVISION REVELATION, let’s look at this student work below:
One student's incorrect ladder division and solving with equal grouping correctly.
The same student's equal grouping and the relationship to the correct ladder division which I showed him. Notice how 20, 30, 10, and 5 are the same in the circles and to the side of the division problem.
So, you ask, what is the big AH-HA? If you notice that the numbers in the student’s circles (equal groups) are the same numbers to the side of the division problem (or the ladder). The numbers to the side of the ladder are actually the chunks that students would naturally break off if they were naturally dividing cookies among people etc.! Now I know according to my PD classes that I shouldn’t have just told and shown the student this relationship, but I should have allowed him to figure it out. I was just a little too excited to hold in my own personal discovery I suppose.
Another discovery I had in experimenting with having students discover their own strategies, I learned that students actually do better and persevere more when the problem is in a context familiar to them and it isn’t just a naked numbers problem. Having a contextual situation actually gives the students an entry point into the problem, and so they don’t give up as quickly. I have always thought backwards. Students MUST UNDERSTAND THE NAKED NUMBERS FIRST and THEN they can SOLVE HARD WORD PROBLEMS. But in reality after my experiment with these fifth graders, they were much better at solving this division problem in their own way when they had a context. I have had a complete paradigm shift, and you were all here to see it above– that is.
My New Singapore Math Manipulatives!
I get the luxury of ordering new math materials for my school. To prepare for all of the common core instruction in place value, fractions, and decimals, I ordered lots of fraction circles, some more unifix cubes, and some place value strips. When my order came in this week, it was like Christmas! That is always the feeling I have anytime I rip open boxes of new math materials! I am most excited about the place value strips which are pictured below. As you can see in the pictures, they come apart so that students actually see the value of the number. These are one of the Singapore Math manipulatives. Each color matches the Singapore math discs if you decide to get those as well which I use with 1st, 2nd, and 3rd grade. If you want to have a smaller size, there is also a copy, cut and laminate version in one of Sandra Chen’s books, The Parent Connection for Singapore Math.
Place Value Strips
Place Value Strips Showing the Value of the Digit
Decimal Strips
Decimal Strips Showing the Value of the Digit
Free Test Taking Rubric or Checklist
I showed the fifth graders that I have been teaching for the past few weeks this page before they tested. I let them know that I was going to be looking for these actions or test taking strategies while they tested. Our principal gave the students extra recess time at the end of the day if they worked hard on the test all morning long. I wanted a way to measure “working hard on the test”, so I used this checklist/rubric. If students did 4 of the 6 actions or testing strategies listed on the sheet, then they were able to have extra recess. Across the top of the page the categories read:
- Underlined Key Words
- Brain Dumped— Writing important information down on the math reference sheet that they may forget
- Eliminated Wrong Answers (on multiple choice)
- Used P.E.C.E (an acronym that stands for using a picture, equation, complete sentence, and elaboration to solve an open response)
- Persevered When Problem Solving
- Checked Work or Used the Entire Time to Work
If you would like to use this form, you can download it for free here. I am posting it in Word format so that you can open it and change the wording to suit your needs.
Testing Rubric Checklist
Do You Need a Fun Way for Students to Experience Perimeter, Area, or Volume?
Students built the following houses out of food items and then calculated the perimeter of them. We did this project as a relaxing activity after testing. I allowed students to build houses out of graham crackers, frosting, red hots, marshmallows, and Smarties. Then they used a measurement tool to calculate the perimeter in centimeters. If I had really wanted to use this activity to stimulate mathematical thinking, I would have had the students calculate the surface area and the volume using large marshmallows. Since I just wanted the kids to take a fun break after testing, I didn’t have them calculate anything except the perimeter. If I had to pick their favorite after testing activity from this week, this would have to be it!
One of the most creative houses the kids built. If you look on the napkin, you can see the perimeter calculations.
One student discovered an array with her Smartie shingles!
Measuring perimeter of the graham cracker house with a base ten rod. (each unit is a cm)
Yummy, Fun, Math with Circles!
I did this today to reward students who had put forth their best effort on our state tests. Originally, I had just planned on bringing cookies, but we had just been studying the different parts of a circle. Cookies being round just naturally lent themselves to be used for this spontaneous activity. I bought some Twizzlers and told students to use those to separate the strands to use for the chord, radius, and diameter. A spoonful of heaping icing was distributed to each child to help all of the parts stick. Students got red hots to use for the center of the circle. According to the common core standards, unfortunately, the parts of a circle will no longer be in the curriculum for fifth graders after I came up with this nifty activity :(. Maybe a teacher somewhere out there that teaches middle school students or older could benefit from this in the future. Below are pictured the circles with their Twizzler radius, diameter, and chord. In reflection, if I were to do this over I wouldn’t have bought Twizzlers. I would have bought those individual licorice colors that are single strands. I saw a package at Walgreens after I had already bought the Twizzlers. I think those would work better because they are already single strands AND I could have differentiated what I wanted students to use for each color. For example, make your chord red, your radius blue, and your diameter yellow licorice. If students did this, then I would really know that they could distinguish between the parts. All in all, the students ATE THIS UP–literally!
A student who followed directions and separated the strands of licorice.
And a student who didn't quite follow directions with the Twizzler strands still stuck together, but recognized the parts of the circle nonetheless. These cookies were a little crumbly from their trip to school. Hence, the imperfect circle shape.
