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!
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:
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.
Students often struggle with labeling the hands correctly on a clock with quarters of an hour. This is for several reasons:
1. Students hear quarter and immediately think 25 cents, so they label the minute hand on the 5 to represent 25 minutes. To redirect students from this misconception remind them that quarter means four parts. Quarts of a gallon means four quarts equal one gallon. A quarter of a dollar means 25 cents because 4 twenty-fives equal 100 cents or a dollar.
2. Students may understand that a quarter is fifteen minutes, but they hear literally, “Show me a quarter after 5.” Students will literally find the 5 on the clock and find fifteen minutes after the five. Then they will put the minute hand on the 8. To correct students explain that it is fifteen minutes after the minute hand hits the new hour or the twelve.
3. Students sometimes are confused with the language a quarter after, a quarter past, a quarter to, a quarter til, a quarter until, a quarter before. To help students with the language, create a chart that shows after and past mean the same things with an arrow showing counter clockwise. Likewise, chart a poster that shows that to, til, until, and before mean the same things with a hand pointing counter clockwise.
Seeing the same problem, students continuing to mix up area and perimeter questions, reoccur with our 4th and 5th grade students on their unit tests, we decided to try something new to help them differentiate between the two. With the questions already cut out, we took all of the released area and perimeter questions from our previous state tests and had students do a sort with them. Pairs of students sorted the questions underneath an area or perimeter heading. To add a little challenge to the activity, we added some volume, capacity, weight, multiplication, and division questions without telling them that these questions weren’t area or perimeter. As teachers, we learned during the students’ sort that students were thinking of area as the space inside of anything so that they were confusing volume and capacity with area. This led to students gaining a deeper understanding of the meaning of area. The students also learned from one another as Bloom’s higher order thinking on the evaluation level was in place. Students had to discuss each question and agree or disagree with one another about the decision to place it underneath a heading. See below for a look at our activity.
The common core standards for first grade state that students must become fluent with adding and subtracting tens. To promote fluency, students need to discover patterns and how they change numbers. Only when students have examined patterns and become comfortable with them should they be given timed “naked number” problems to assess and improve their efficiency in recalling the patterns. Finding the answer to a “naked numbers” problem needs to be merely a by product of knowing the pattern.
When trying to figure out how to teach fluency with adding and subtracting multiples of ten to first graders, I struggled to find resources to do this. Because of the lack of resources, I developed these sheets for next week to lead students and their teachers into guiding discussions about patterns on the hundreds chart. They are simplistic activity sheets but necessary. I will be trying them out this week. I am providing a link to them below. If you would like to try out one of the lesson sheets, the preview file will give you a sample for free at TPT.
I recently attended a fabulous professional development which discussed the Eight Mathematical Practices for Common Core Standards. One of these practices is having students reason abstractly and quantitatively. I have always required students to label the answers to their equations when supporting an open ended question, but I didn’t realize the depth of what I was requiring. Consider this. Which is more– 5 or 8? We would all agree that 5<8. This is an example of reasoning abstractly. Now, consider this. Which is more–5 quarters or 8 pennies? We would all agree that 5 quarters > than 8 pennies. This is an example of reasoning quantitatively. When students answer a word problem by labeling each component of their equation, they are reasoning quantitatively. When students merely solve a “naked numbers” problem without labeling the quantities with which they are reasoning , then they are reasoning abstractly. When students are reasoning with specific quantities, then the foundation being built is stronger for them to then build abstractly. Just think. The math textbooks are backwards. Quantitative reasoning problems follow abstract reasoning, however students understanding is built first with quantitative problems and should be followed with abstract understanding.
To practice math facts, spelling words, or any other quick answer type learning, you can play Squat. To play Squat, two students from two different teams approach the board. The teacher calls out a fact or a spelling word. The two students at the board race to answer the question correctly and then they squat when they think they have the correct answer. If they are correct they earn a point for their team.
When I have played this, I usually split my class into two teams. Different students on the teams take turns to be at the board to earn their team points. Team points can be taken away from students who aren’t waiting quietly or who blurt out an answer when it isn’t their turn. Students love this game and will beg to play it after you have played once. If you have some extra time (heh, heh, who has that?!) during a spot in your day, this is a fun way to reinforce skills or fill time.
Another math coach related to me today the story of how a student she taught had named fingers sections as something that comes in groups of threes. She took this concept and helped students use this to develop multiplication strategies to learn their threes multiplication tables. Fours multiplication tables can be learned as well if students include counting the top part of their palm. See the pictures below for more clarification.