Physical Models of SubtractionCarrying on from out last week's 'AHA' moment, where we realized we needed to support our students understanding of the concept of subtraction, even if they could already use the algorithm (how we were taught to subtract - or the stacking method as the students call it). From the book, Solving for Why: Understanding, Assessing, and Teaching Students Who Struggle With Math, by John Tapper, we know that we want our students to develop a broad conceptual understanding of mathematics as well as a procedural understanding. "Mathematical models are mental constructions that we all use to understand complex ideas (Schoenfeld 1994), Without models on which to ground our mathematical thinking, we may only attain the kind of knowledge that comes from practicing every kind of mathematical procedure we ever need. This understanding is not flexible; rather, it is bound to particular contexts. We can see this with our students when they are only able to solve problems that are matches for the instruction they've just received. They can, for example, solve problems with multi-digit multiplication, but only after they've just practiced it. Without a model to ground mathematical understanding, students are unlikely to generalize what they know (Ryan and Williams, 2007)." from Tapper, J. 2012. Solving For Why. Math Solutions; Sausalito, CA. As we explored physical models to represent the subtraction situation, 82 - 57, we realized that some models were just re-writing the numbers using manipulatives, for example playing cards. So, we worked on ways to represent the actual number of objects. We then realized it was important to organize our objects so that people could easily see the numbers we were subtracting. Various strategies were developed including; Grouping by 5s or 10s, colour-coding, using other manipulatives as dividers. Our next challenge was for students to use the manipulatives to show the regrouping required to solve this problem. A couple of groups demonstrated their thinking by using colour-coding and a series of steps. They were able to verbally explain what was happening in each step of the subtraction process. This group explained their process on video. Our next step was to have students demonstrate their thinking in a different context; by using representational or drawn models of the subtraction process. Several models showed the students clearly understood and could visually represent the process of subtracting 26 from 53.
0 Comments
Satisfying Story EndingsWe started by asking the students if they'd ever read a book or watched a movie that they hadn't like the ending of and why. Some of their responses included: - it was predictable - you knew what was going to happen throughout - it was left as a cliff-hanger - you wanted to know what would happen next - the problem was solved but then it just kept going; another event was added, and another, and another, and.. - the ending doesn't connect to the story - everyone died or everyone moved away - "They woke up. It had all been a dream." - "They lived happily ever after." -"To be continued..." Students looked at different picture books, for ideas of what makes a satisfy story ending and have been exploring Barbara Mariconda's "Extended Ending" techniques as one way to create satisfying endings. Her idea is that in most narrative stories, the character usually experiences something significant - they have solved a problem, had an adventure, or been involved in a significant experience of some kind. As a result of their experience the character has changed in some way. An extended ending reveals what the story is about and how the main character has grown or changed as a result of their experience. It has a natural sense of closure. Techniques that can be used to create a satisfying or extending include: Decision - something the character decided to do as a result of their experience in the story. Wish or hope - something that the character remembers most about the main event. Feeling - how the main character feels about what happened in the story. Memory - the main character reflects back and remembers something about the main event. After several opportunities to identify these techniques in extended story endings, students are working together to develop extended endings for "How to Catch a Star" by Oliver Jeffers. Here are some of our initial, unedited attempts: "The boy remembered how lonely he was when he didn’t have a star. That was then. Now he had a star, right then the boy made a decision he decided it would always be his star. The boy hoped the star would never leave him, and they would be friends forever." ~Claire M~ "He decided one star is enough, for now. He hoped the star would be his friend forever. He felt amazing when he finally touched the star’s delicate body. He thought back to when he longed for a star and now he had one of his very own." ~ Payton, Saviero, Claire ~ "He remembered how he was so sad and lonely when he didn’t have a star but now he hoped that the star would play with him. If not he would decided he would not be sad." ~Brooke, Jamie, Adam ~ "He remembered when he was trying to catch a star. He remembered the joyfulness, the loneliness, and the sadness, So he decided that he always wanted do challenging things." ~Clive, Ryan, Lachlan~ Subtraction StrategiesWhen we asked our students, as a pre-assessment, to show us all of the subtraction strategies they could find for a given problem, we noticed some patterns to their thinking. Some students were able to represent their thinking using tallies. Some tried this strategy but quickly tired of the process of drawing so many tallies. Some students drew pictures of their thinking and used a "code" such as colours to represent the different amounts. But most of our students, went right to the numbers and used various number strategies to develop a solution. Some examples include: We know how important using manipulatives and making models are to developing a conceptual understanding of mathematics so we embarked on a journey to have our students represent the concept of subtraction with models. Check back next week for images of their models.
Elk Interaction Picture In our continued look into 'How Does Human Interaction Change and Identity,' the kids were given a photograph of a group of people standing near an elk. They were asked to consider all of our previous learning including the wildlife interaction rules that they had created. We decided to assume that all of the characters in the picture, including the baby and the elk, new the rules. The thought bubbles that the kids created show what each character might be thinking about when being so close to a wild animal. NumeracyWe continue to develop numeracy strategies through exploring different ways to add by developing reasoning skills through strategy games, and working towards clear explanations of our mathematical thinking. Front end addition is a useful strategy that can be used to add 3-digit numbers. In this strategy, and these examples, numbers are partitioned into hundreds, tens, and ones and added. This is called front end addition, because you start with the front end of the numbers or the hundreds. Students find most success with this strategy when they carefully line up their addition into columns as would be found on the place value chart, either by physically drawing lines, or by carefully lining up their numbers as they record them. Closest to 100 or Closest to 1000 Students have working with a partner, using two decks of playing cards to develop estimation and mathematical reasoning skills. If playing to 100, students draw 6 cards and choose 4; if playing to 1000, students draw 8 cards and choose 6. With the cards they have chosen, students combine their cards to make 2 two-digit numbers (Close to 100) or 2 three-digit numbers (Close to 1000). So What Are the Best Numbers To Make? When mathematicians prove their thinking, they often use a similar problem with smaller numbers to explain their thinking. After playing this game over a few days, we wanted to get a sense of our students strategies and reasoning. We provided the students with a situation in which they had flipped over these 6 cards when playing Closest to 100 6, 3, 1, 4, 8, 7 They were asked to make the best number possible, just like they would do if they were playing the game. Some of their thinking and strategies are listed below. Trial and Error: Clive's notebook demonstrating his process of testing possibilities. Trial and Error and Thinking About Other Possibilities: The best number you could make would be 13 + 87. I looked at a few and they weren't far from 100 so I knew that I could find a better answer. I looked over the numbers and saw that some were closer. I found 87 and 14 and saw that was 101 so it was pretty close. Then I found 87 and 13 and that was 100 so I knew it was the best number. There might be others that make 100 also. ~ Finn~ Trial and Error, with evidence of thinking: "61 and 48 because when you add them up they make 109 and its only 9 away from 100. It's an Ok answer but not the best. I was too big, so next time I would choose 47 and 63 ... oops, that's too much. 63 and 37... I looked at 47 and decided that I needed smaller numbers. This time I got 100." ~Adam~ Comparing sizes: I think 13 + 87 because 13 is way lower than all of the other numbers and also 87 is really close to 100. If you add them up then you get 100. If you split them into expanded form then you would have 80 + 7 and 10 + 3. Then you add 3 + 7 = 10, then 80 + 1- = 90 then add 90 + 10 = 100. Expanded form is a great strategy that helps me a lot. ~ Chloe~ Considering Probability: 87 and 13 would be the best numbers because 87 + 13 + 100 and we are going for Closest to 100 and my answer is exactly 100 and if I was playing against someone I have a high likelihood of winning. ~ Mayan~ 86 and 14 = 100 I decided to keep this number because it is either equal to the others or better than the others. ~Georgia~ Developing a Strategy (although not all of the following responses resulted in the best solution at this time, these strategies demonstrate examples of strong student reasoning throughout our investigation): I would put 8 and 1 in the tens because it make at 90 and I would be close. ~Saviero~ I thought 80 + 10 = 90 so I knew it was close. If I had 2 large numbers I knew it would be waaay too high so I chose 18 ... a small number. ~ Lachlan~ 14 and 86 ... These are the best numbers because we're trying to get to 100 and all the other cards are too high or too low. ~ Payton~ I chose 13 and 78 because I knew that if I added 80 with 10 I would be getting close. If I added 10 more from the ones I'd be there.... I wanted to get close to 100 but not over. ~Sofie~ 13 and 87 because the rest of the numbers are not as close to 100 like 63 + 18 = 99 but you can get 100 a different way 83 + 17 = 100. ~Jamie~ Addition Strategies - Revisited The following example demonstrates a sound strategy for the game - finding ways to combine two cards that equal 90 and then finding two additional cards that total to 10, so that the sum will be 100. On the surface, this student appears to understand addition using the traditional algorithm, the strategy most of us as adults were taught in school. By observing the student's explanation, we notice misconceptions about number that may be fostered using this strategy. The student does not articulate that they are, in fact, adding 80 and 10 together to get 90. This becomes problematic for some students, who when taught this strategy, do not fully understand the place value inherent in addition. I looked for what equals 9. 8 + 1 equals nine. Now I need something that equals 10. I could do 6 + 4 = 10. (This student is really looking for what equals 90) Both expanded form addition (see examples on our last blog post) and front end addition (examples above) are addition strategies that support students with developing a strong understanding of the place value of addition. Students who are fluent with using either of these strategies may not use the traditional algorithm and are often quicker at solving these addition situations mentally. |
AuthorsMrs. Montanaro, Mr. Messer and Mrs. Austman teach grades 3/4 at Elbow Park School in Calgary AB. Archives
April 2017
Categories |