Journal Summaries 2

Journal Summaries 2

Journal Summary #1

Article: Counting On Using a Number Game

           

-NCTM, Teaching Children Mathematics, March 2015, Volume 21, No. 7 (p. 430-436)

Summary:

            This article discusses how to help students who use the counting-all strategy for adding quantities switch to using the counting on strategy.  These strategies can be used, for example, when counting the total number of objects in two sets. While these strategies are similar, there is a clear difference. Take a look at the following example: There are four objects and two more objects. If counting all were used, children would count the first set of objects (“one, two, three, four”) and then move to the other set and continue (“five, six”). Taking that same example, a child who was using counting on would not need to count both sets of objects. They know that one set has four objects so then they continue to count on with the second set of objects (“five, six”). In other words, children who use counting all, use the objects in both sets to figure out the total number of objects whereas children who use counting on do not need to use the physical objects to figure out the total number in both sets (they just use the counting sequence to figure out which number comes next; so in the previous example, they know the first set of objects has 4 and then they think about the next number in the counting sequence and add on, so five, six). When using counting on, students know the quantity in the first set of objects and then they may figure out the quantity in the second set by using their fingers, bobbing their head, verbalization cues, etc.

Why is counting on more beneficial for students? Counting on is a strategy that many teachers try to implement in their students because students who use this strategy have a better number sense. They already know the quantity in the first set and are simply looking at the second set and adding on to that first set. This means that students who are using counting on are able to decompose. Unlike students who count all and only see one set of objects, students who count on see one set of objects and see that it can be decomposed into two sets. This decomposition helps students master basic math facts. Many young elementary students who have difficulty in the math area are using the counting all strategy when they should be using the counting on strategy. However, it’s not that simple of a job for teachers to get students to switch from counting all to counting on.

            Two approaches that can be used to help students count on are: the Make Ten card game and having the teacher modeling counting on using playing cards. Despite those helpful strategies, many students continue to struggle with counting on. Therefore, there was a game board developed by Siegler and Ramani that is a great tool to use to help students count on. There is a standard hundred chart (but this could be adapted; so if the teacher chooses they could have a 1-50 number chart, for example). Students would roll a die or spin a spinner. The number that they get after rolling a die or spinning a spinner will determine how many spaces they move their token (the token starts at number 1). As each student moves their token, they have to say every number that their token touches. For instance, let’s say a student’s token is on the number five and then that student rolls a 4. Instead of counting, “one, two, three…” as the student moves the token across the spaces, the student would move their token so it touches each space and say “six, seven, eight, nine.” In other words, as they move their token one space, they would say “six,” as they move it another space, they would say “seven,” etc. The student needs to find an alternate way to determine that they moved four spaces. Therefore, while the student is naming each number their token touches, they also have to determine how many spaces their token moved (they can use their fingers or head bobbing, for example, to track how many total spaces their token moved). For example, the article described an elementary student named Tina. Tina rolled a 4 and bobbed her head to move her token. She bobbed her head twice, paused, and then bobbed her head two more times. She then realized that 2+2=4 (this is a decomposition strategy; she decomposed 4). This article describes a teacher using this game board as an intervention strategy to help four 2^nd or 3^rd grade children who were struggling with mathematics. They played once a week for twenty minutes weekly. By the end of the four weeks, three out of the four children became more proficient using the counting on strategy. The more students play this game/practice with counting on, the more likely they will gain a better understanding for math facts and start to do mental computations. Teachers need to model the counting on strategy while also letting students explore and discover their own strategies.

My Thoughts/How Useful it would be for a Classroom Teacher:

            Before reading this article, I never even thought about counting all vs. counting on. I also obviously did not understand the difference between these two strategies. However, after reading this article, I understand so much more about the way students think about counting as well as how they should be counting (I feel that many students use the counting all strategy when the counting on strategy should be used). I believe that there are so many valuable aspects to having students counting on as opposed to counting all, and I could envision how this strategy can be effective for elementary students. I think students can truly gain a better number sense and math fact knowledge. While I do believe counting on is a great strategy and the use of the game board can help teach this strategy, I do think it may be confusing for students because they have to simultaneously think about counting each number and the spaces they moved. The article even mentioned one student (who likely has other learning needs), couldn’t catch on to this counting on strategy even with the help of the game board. Overall, I do think this counting on strategy is useful for teachers in a classroom because students can be more comfortable with counting, cardinality, and sequencing. By modeling counting on and having students explore counting on, they are required to think about the sequence of numbers, more vs. less (ex. realize that 4 is bigger than 3 and comes after 3), and the decomposition of numbers (ex. 4 can be broken down into 2 and 2). If teachers introduce counting on and the game board, I think students will be engaged as well as grasp and retain more math facts.

Journal Summary #2

Article: Assessing for Learning

           

-NCTM, Mathematics Teaching in the Middle School, March 2015, Volume 20, No. 7 (p. 424-433)

Summary:

            Teachers need to constantly assess their students’ understanding, especially to determine which students are excelling (and may need to be more challenged) and which students are still struggling. With the fairly new Common Core State Standards for Mathematics, students are expected to know, understand, and achieve a better mathematical understanding. Teachers need to assess students to make sure they meet the CCSSM goals as well as to make sure they understand the math content. This article discussed how classroom teachers could design better assessments and classroom practice. First, I want to point out that formative assessments are those such as warm-ups, questioning, class discussions, etc., and summative assessments are those such as tests, projects, and quizzes (which determine grades).

            This article discussed how student learning can be broken into cognitive types or learning levels (Cangelosi’s learning levels): 1. Construct a concept, 2. Discover a relationship, 3. Simple knowledge, 4. Comprehension and communication, 5. Algorithmic skill, 6. Application, and 7. Creative Thinking. These levels help students learn/think about mathematics and are ordered logically (it’s a learning progression). However, this order can be switched around (not confined to this specific order). This article also mentions about scoring guides vs. rubrics, which is important to note. A scoring guide is quick to grade but doesn’t provide feedback for the students whereas a rubric takes longer to construct but it makes grading easier and gives the student more feedback. The article discusses assessment item design strategies based on these learning levels in a seventh grade math classroom pertaining to content about circles (area and circumference of circles).

            For each learning level, the teacher of this seventh grade level may assess in the following ways: 1. Construct a concept: Give students prompts that have them sort or categorize to show they have built a concept in their mind. Make these prompts connected to students’ lives. The teacher could also ask students to describe a concept’s characteristics and/or show examples or nonexamples. 2. Discover a relationship: Students can report what they learned through narratives (can state facts they learned and relationships they found). 3. Simple knowledge: Have students respond to a question based on what they remember (ex. state a formula or name a method used by mathematicians). 4. Comprehension and communication: Give a prompt where students need to explain (literal and/or interpretive understanding of a math expression, process/method, or message). This is almost like a novel for students where they need to explain their understanding to prove that they aren’t just memorizing facts. 5. Algorithmic skill: Students recall a sequence of steps and show a math procedure (ex. State the answer and have the students show the process of how to get to that answer). 6. Application: Students are able to skillfully decide how to solve the problem. No clue words should be involved in these problems, because at this level, students should know what they need to do. 7. Creative thinking: Design tests so students can show their creativity (art can be included but is not a necessity). Using synectics (connecting seemingly unrelated ideas) and open-ended application prompts is recommended. Students who demonstrate unique, different, and out-of-the-box (originality) math thinking will get a higher score. Many students are not comfortable with creative thinking because they don’t get enough practice with it, but after students practice creative thinking/exploration exercises, then these creative problems can be incorporated on summative tests.

My Thoughts/How Useful it would be for a Classroom Teacher:

I am a firm believer that a test shouldn’t just be a test. In other words, a test shouldn’t just be created to give students a grade. Also, tests shouldn’t just be used to evaluate how well teachers and schools are performing. Tests should be a tool that helps the students; they should be carefully constructed so students can demonstrate their understanding and learn something from the test. They should also be used as a way for teachers to determine where students are still struggling so he/or she could help the students with these struggling areas. I think using Cangelosi’s learning levels is a great strategy to ensure that students are getting a variety of rich learning problems/situations that deepens there mathematical knowledge and understanding.  However, I do think that some teachers may be too lazy to take the time to tie Cangelosi’s learning levels and standards to an activity. A lot of careful thought and planning has to go into strategically structuring a problem so that it meets the standards and Cangelosi’s learning levels. I think his levels are good because students are really exploring the content in multiple ways and slowly progressing towards more challenging/creative thinking. As previously mentioned, though, I’m not sure if teachers will realistically employ these levels. Overall, I do think his learning levels are useful for math teachers in a classroom because students can work with more basic, concrete knowledge problems and slowly be introduced to more critical thinking, creative, and abstract problems (building upon their foundational knowledge). I think of these cognition levels almost like a staircase. With each stair, the level becomes a little more challenging and requires the students to do a little more thinking than the step before.

Video Analysis 2: "Number Operations: Multiplication & Division"

“Number Operations: Multiplication & Division” Video-4^th Grade

This was a math video called “Number Operations: Multiplication & Division.” Becca Sherman was conducting lessons in a 4^th grade classroom about multiplication and division. These clips, much like the video clips we had to watch from the first video analysis, were valuable to watch because it was as if I was able to jump into these students’ minds and see their thought process. I was also able to observe the different teaching strategies the teacher used (i.e. the bell to gain students’ attention, partner sharing, etc.). Additionally, it was interesting to see the various math approaches the teacher exhibited (i.e. bar model, array of dots, etc.) and how well the students were able to comprehend these approaches. There were no video clips of the teachers discussing pre-planning or the goals of the lessons. The lesson simply began with the teacher introducing the lesson. There was an introduction (Part A and B) presented to the students, problems (the lessons) that were split into four parts (Problem 1-Part A and B, Problem 2-Part A and B, Problem 3-Part A and B, and Problem 4-Part A, B, C, D, E, and F), and a closure (then followed up by a faculty debriefing video clip).

Introduction:

           

            Ms. Sherman made the students aware that they were working with multiplication and division. She also pointed out the quote that was written on the board, “A picture is worth a thousand words.” After asking students what they thought this quote meant, she indicated that they will try to get to know what that quote means after practicing with today’s lesson, multiplication and division. Ms. Sherman next asked the students to think about something they know about multiplication. They then shared their thoughts about multiplication with a partner. It was interesting to watch the “Introduction-Part B” video clip of the male student trying to define multiplication. It was obvious that he struggled to devise a definition of multiplication. He may have known how to do multiplication (knows examples; 2x2=4) but doesn’t know what it actually means to multiply, as I think is the case with many students. Or maybe he didn’t really understand any aspect of multiplication and therefore, was completely confused all together. I think a common misconception when teaching mathematics is that many educators do not ask “why” enough and/or assume their students actually know what the meaning is behind what they are doing. However, if students and teachers do not explore the “why,” the students aren’t going to be able to truly gain an understanding and make sense of the concept they are working with. This, in turn, results to students likely confronting more struggles with the math concept and less success.

The Lesson:

Problem 1-Part A and B & Problem 2-Part A and B

           

            In the video clips, Problem 1-Part A and B and Problem 2-Part A and B, the students were accessing their prior knowledge about multiplication and division. The teacher was able to assess their understanding and confidence levels about those math concepts. Ms. Sherman had “Multiplication” and “Division” written on the whiteboard. In Problem 1, students were looking at the term “Multiplication.” After sharing thoughts about multiplication with their partners, responses were shared as a whole class. Some students agreed that multiplication was like adding (i.e. 2x8=16 is the same as 8+8). Other students provided similar examples (i.e. 5x2=10 is the same as 5+5). Students seemed to be stuck on timing by 2’s and struggled to think of other examples (Is what you are saying true with times something else?). The students brought up the idea of “groups” (i.e. 9x2 is the same thing as two groups of nine and 9+9). Ms. Sherman then emphasized this idea of “groups” saying that they needed to create equal groups and could use pictures to help them create these groups. She drew a bar model of two groups of 3 and had a student come up to the whiteboard to add another group of three to the bar model.

After discussing multiplication with partners and the whole class, students were asked to discuss “division” with their partner (What do they know about division?). After sharing thoughts about division with their partners, responses were shared as a whole class. Some students believed division was like subtraction while other students believed division was like multiplication (i.e. “It’s like multiplication but you divide it. Like 9 divided by 3 equals 3 and 3x3=9; “It’s like multiplication but you switch it. Like 5x3=15 and 15 divided by 3 equals 5.”) Ms. Sherman asked the students to elaborate on their thoughts about whether they thought division was like subtracting or multiplication. After briefly drawing a fact family triangle (which the students seemed to be familiar with), Ms. Sherman was trying to get the students to aim their responses towards the idea of equal groups. She threw the number “100” on the whiteboard and wanted the students to think about how they could divide it into some equal groups. One student mentioned that 100 could be divided into two equal groups of 50 (100 divided by 50 equals 2 and 2x50=100). I think Ms. Sherman realized that she probably started with too large of a number for the students to break into groups, and therefore, she decided to have the students work with the number 12 (instead of 100). She drew 12 dots on the board (a 3 by 4 array) and wanted students to try and make equal groups out of that. Many students were able to successfully make equal groups (i.e. 3 groups of 4, 4 groups of 3, 2 groups of 6). I think many students were starting to see the connection between multiplication and division.

 Problem 3-Part A and B, and Problem 4-Part A, B, C, D, E, and F

            For the next problem, Problem 3, students were to use mental math to think about the problem 26x4. I think Ms. Sherman wanted to see if students could apply the bar model and the idea of grouping. After students independently thought about this problem, there was a partner discussion and then a whole class discussion. Before discussing their methods, Ms. Sherman first had students share their answers they got from solving 26x4 mentally. One student mentioned they got 104 as their answer and another student mentioned he got 70 as his answer. Ms. Sherman also asked the students if this problem had only one correct answer or multiple correct answers, and many students said multiple correct answers (which was surprising). Ms. Sherman made sure to clarify that students knew that this problem only had one answer. It seemed many students did the traditional stacking multiplication strategy where the problem was set up vertically, and the students multiplied and “carried.” Next, Ms. Sherman wanted students to indicate how they could draw a picture of 26x4 (since "a picture is worth a thousand words"). She drew a bar model with 4 groups of 26.  One student shared that he did 26+26=52 and 26+26=52 and then added 52+52 (so he doubled 26 and then doubled it again). Lastly, another student shared a different approach. This student said, “I knew the 20 was in the 10s place so I did 20, 40, 60, 80 four times (20 times 4). Then did 6 four times, 6, 12, 18, 24 (6 times 4). Then got 80 and 24, added it, and got 104 (the correct answer)." I thought this was very advanced thinking and was surprised to see that this student thought of the problem in this way. Technically, this student broke apart 26, and he broke it into a 20 and a 6 (factor tree). I believe many students were trying to figure out different methods as to how to solve the problem, which was great, but they still seemed to struggle with grouping/bar model.

            The last problem that the students worked with, Problem 4, asked the students to take the words of a story and make a math picture to help solve the problem (now going from word form to picture form). The problem was as follows: “Maria saved $24. She saved 3 times as much as Wayne.” This problem was different in that no questions were posed. Students were asked to think about this story, what they know/what’s going on, and who is in it. The students then independently thought about this problem, drew a math picture at their desk of this problem, and then shared what they drew with their table. It was interesting to observe that at first, many students struggled drawing a picture to represent the problem. As they kept working at it, many students eventually expanded their numbers that they wrote down into more of a picture form. However, they still faced trouble when trying to explain what they wrote/drew. Furthermore, it was remarkable to see how split up the class was on the problem; about half the class thought $72 was the answer, and the other half thought $8 was the answer. This showed me that while many students can solve multiplication and division problems in number form, they have a misconception when reading story problems and deciding what concept to apply (multiplication or division). In other words, many students saw the word “times” in the story problem and assumed that meant to multiply (24x3=72). 

            Ms. Sherman then introduces a student’s strategy, Charlie’s model (bar model). His model showed Maria’s money with three boxes that totaled $24 and showed Wayne’s money with a question mark (need to figure out how much money Wayne has). These four boxes were all in green showing that they all have to be the same number (equal groups have to go in the boxes). The last question mark is the total of all their money. On a piece of paper, students copied Charlie’s model and filled in the missing numbers. Similar as to how students thought before Charlie’s model was introduced, many students either put 24 in the boxes or put 8 in the boxes. After students worked with Charlie’s model at their desks, Ms. Sherman presented them with two questions: How much did Wayne save? How much did Maria and Wayne save together? The students were then instructed to answer these questions underneath their pictures (needed to think about if there was a number that would be the same for all the green boxes that would work for this story). When the whole class discussed Charlie’s model, a student suggested putting 24 in Maria’s boxes. Ms. Sherman explained why 24 couldn’t be possible (Maria doesn’t save $72; she saves $24). Another student shared that Charlie put 8 in all the boxes (because 8x3=24 so three boxes of eight shows that in all Maria has $24). I don’t think many students wrapped their head around Charlie’s model. I believe many students had difficulty understanding why $72 was not correct as well as had trouble understanding why it should be $8 in all the boxes and why the total was $32.

Closure

            Ms. Sherman led students in the idea of putting $8 in all the boxes and showed the students that if they have 4 equal groups of $8 (4x8), they have $32 in total. The last task Ms. Sherman had her students participate in was to write approximately three sentences at the bottom of their paper about what they will remember about Charlie’s way (what they liked, didn’t like, something about his drawing, if it made sense to them, etc.). I think many students struggled to reflect on Charlie’s model and how he solved the problem.

Faculty Debriefing:

           

            Ms. Sherman was glad that the students were so eager to share their thoughts about multiplication and division. For multiplication, Ms. Sherman expressed to the other faculty members that it was hard to get the students to expand with the 2’s. Moreover, she expressed that the idea of equal groups didn’t come up naturally (she had to introduce and emphasize the idea of equal groups). Additionally, Ms. Sherman mentioned that the students said that multiplication is like addition and division is like subtraction. However, they couldn’t back this up (couldn’t explain this). Furthermore, Ms. Sherman decided to extend the warm-up longer so students could try to explore the idea of groups when dividing. However, she still felt she was almost forcing this idea upon them. When students were doing their drawings of the problem during the lesson, Ms. Sherman was happy that the students were engaged and thinking. She also appreciated that even though the questions weren’t presented at first, the students were starting to think of answers to the questions. Ms. Sherman was well aware that many students had the misconception of 24 x3, and when she put up the Charlie model students still struggled with the misconception of 24 x 3. She said that half of the students incorrectly wrote 24x3 to find out that Wayne had $72 (which was three times as much). This proves that despite her explanations of grouping and her pictures she displayed, many students still struggled to grasp the concept.

            Another faculty member shared with the group that he saw that a student had $8 for his answer but when everyone else at his group said that they had $24 for their answer, the boy switched his answer to $24 and the boy was unable to explain why he switched his answer. The idea of copying other students’ answers presents the important question, “How many children understood it and how many had copied the correct answer?” Other faculty members noticed that students struggled to explain their answers and/or did not understand what number was correct to put in the boxes.

            I think the faculty members realized the importance of getting students to understand multiplication and division and the connections to equal groups rather than just understanding math facts (students need to understand the “why”). As opposed to simply throwing out math problems/facts, students need to first build their algebraic and mathematical vocabulary to help them gain an understanding and deeper confidence level for the math concept they are working with.

Overall Thoughts:

            Watching these video clips was a great learning experience and provided me with some positive and negative thoughts. First, I loved that the students were so engaged in the lesson and that lots of questions were being asked to stimulate student thinking. Additionally, I liked how she asked one of the students to come up to the whiteboard and draw her answer. I also liked how students thought independently then shared with a partner and then shared with the whole-class. Furthermore, I enjoyed how Ms. Sherman used the bell to grab student attention, and also how she would say something like, “Show me you’re ready for mental math by putting your hands on your head.” I think these are great attention grabbers that cue the students in and regain their focus. I also enjoyed how Ms. Sherman took the time to write what the students said on the anchor charts (she didn’t just discuss their answers, but she also wrote them down). This provided students a nice visual as to what their peers were saying. While there were aspects I appreciated about the video clips, there are various negative aspects I would like to point out. First, while she guided and engaged the students by asking a lot of questions, I think she would ask too many questions at once. Many times Ms. Sherman would ask a question and then right away ask another question right after the one she just asked. I think this confused students and didn’t allow them enough process time. I think it would have been more beneficial if she asked one question at a time as well as provided enough wait time before saying anything else. Additionally, I liked the quote she displayed on the board (“A picture is worth a thousand words”). However, she never came back to this quote at the end. I think it would have been more meaningful if the students came back to this quote and discussed how this quote relates to multiplication and division. Similarly, Ms. Sherman often said throughout the lesson, “We’ll come back to this idea/or this picture later,” but she never did go back to it. Often times she would skip around, and it was frustrating because it was hard to follow her. If I were her student, I would have been overwhelmed and probably would have lost focus. Speaking of focus, I noticed a lot of students started losing focus towards the end of the lesson. I have a feeling that many students were so lost near the end that they just stopped paying attention. Sometimes I felt that Ms. Sherman was trying to force too many ideas upon her students at once, which caused for too much confusion, stress, and frustration. Maybe Ms. Sherman needed to split the days (one day for multiplication and another for division). I also think she needed to spend more time on developing vocabulary to lay a better foundation for this lesson. While I think it is beyond meaningful to let the students explore for themselves, the students also need to have the right amount of preparation for this exploration. Ms. Sherman had to clear up so much confusion in this lesson and asked so many guiding questions that the open exploration may have hurt more than helped. Once again, maybe if she worked on multiplication and division and built some vocabulary knowledge before today’s lesson, students may have been more successfully prepared to explore.

            I enjoyed watching these video clips, and not only did I learn some good tips to implement into my future classroom, but I also learned some strategies NOT to implement. I learned that it’s really important to think about your classroom, if your classroom is prepared for your teaching material, how to adapt your lessons to best meet the needs of your learners, and what tools/strategies you can use to help your learners to succeed the most. Hopefully, Ms. Sherman will go back and continue discussing multiplication and division with her students and build upon the vocabulary, because while they have a good introduction to these concepts, they could definitely use some more practice!

NAEP Reflection

          The NAEP project was a great experience. I realized the value of providing further instruction (curricular implications) so students could continue to practice the skill to avoid the same mistakes. I also recognized the importance of creating clear, well-developed, logical, and effective rubrics when grading student work. Sarah, Kaitlin, and I reviewed a 4^th grade math problem (Graph of Pockets) where students analyzed and interpreted bar graphs. We evaluated students’ answers and used a rubric to score their answers. I thought this process would be fairly cut and dry. However, once we started looking at the students’ answers and comparing them to the rubric, it was much more of a challenge than I anticipated. I think we struggled with so many students because they seemed to fit into more than one category (i.e. some students’ answers could fit into both partial and minimal). We may not have scored all of these answers perfectly, but we did remain as consistent with our grading as possible. Each person may interpret the rubric differently, but the key was making sure we followed our interpretation and graded consistently. I learned that rubrics are a great grading tool. However, the creation of the rubric can make or break how easy it is to grade the students. Rubrics need to be written with careful thought, making sure each level has clear differences. I learned about the making of rubrics in some of my other methods classes, and despite the format of the rubric (i.e. 4, 3, 2, 1, proficient, satisfactory, unsatisfactory, etc.), each category must be different so it can be easily determined where the student falls. After picking three of our seven students (one extended, one partial, and one off task), we created a learning activity where all students could participate. It was definitely hard to think of an activity and make sure it was tailored to each student. However, in a classroom there will always be students who have different levels of understanding, and it is critical that each student’s needs are being met. For our activity, we created a human bar graph/SmartBoard graph where all types of learners could be successful and appropriately challenged.  Throughout this NAEP project, I gained a lot more knowledge about how to continue practicing the content/providing further instruction, adapting activities to support all types of learners, and creating clear rubrics.

            I thought it was great when we shared our NAEP problems as a class. It was interesting to see how the other groups graded their students and the struggles they encountered. It seemed that each group developed a certain system when grading; each group stuck close to what the rubric said and were consistent with how they graded. I was glad to know that my group wasn’t the only one to struggle with the rubric! Some students seemed to be in between scores and all groups were challenged by this problem. The 3rd-5th grade group (tile problem) had an especially hard rubric! It was confusing and hard to scroll up and down between the rubric and appropriate answers. I think it’s safe to say we all learned the importance of developing clear rubrics with easily distinguishable categories/criteria!

Math Applets/App Review

Below is a review of two applets and one app that I found beneficial for students in grades K-2, specifically focusing on number system and rational numbers (counting, cardinality, number and operation in Base 10):

Applet #1: Grouping and Grazing

Source:

            -From Illuminations: Resources for Teaching Math

URL: http://illuminations.nctm.org/Activity.aspx?id=3526

Summary:

          This applet is intended for grades Pre-K-2 and the math content is number and operations specifically focusing on counting, adding, and subtracting. Students learn about grouping, tallying, and place value. This applet has an arrow in which you can select a category. There are four categories on it to choose from: 1. Count by 5s, 2. Count by 10s, 3. Add, or 4. Subtract. When a student clicks each category, the directions are given and the applet orally reads the directions as well. In the count by 5s and count by 10s categories, the screen displays green grass that is divided into two parts (a top half and a bottom half). For the count by 5s category, there are a certain number of cows displayed on the bottom half of the grass. Students have to hit the “select” button and count 5 cows and then hit the “group” button. Once the student presses the “group” button, a space ship will then automatically lift the 5 cows and group them together on the top half of the grass. The students continue this selection and grouping process until they cannot make any more groups of 5s. Any cows that are leftover, the student must drag to the top half of the grass. As the students make groups of 5’s and as they drag the leftover cows to the top, a tally chart is shown keeping track of how many cows there are. Once all the cows are dragged to the top half of the grass, the student presses the “check” button. Then a box comes up that says, “How many cows are in the field?” and the student can either press, “count with me” or, “count alone.” When they press, “count with me,” the applet counts the total number of cows aloud with the student as the student clicks the groups of cows and leftover cows. When they press, “count alone,” a calculator comes up and students click the number of cows they think there are and then they press the check button. The applet verifies this amount. If the amount is too much, the applet will tell you it’s too much. Likewise, if the amount is not enough, the applet will tell you it’s not enough.  Once the student correctly answers the problem, they are done with the problem. They then click “new set,” and another problem is displayed with a different number of cows. For the count by 10s category, the same process occurs except students have to group the cows by groups of 10 rather than 5s. The addition and subtraction categories are slightly different. In the addition category, there are two halves, but the bottom half is split in half again. An addition problem is displayed on the bottom half. For example, the problem 16 +6 may be displayed. The first part of the addition problem is on one bottom half and the second part of the addition problem is on the other bottom half. Taking the 16+6 example, 16 of the cows would be displayed on one bottom half and 6 of the cows would be displayed on the other bottom half. The student has to hit the “select” button and bring all of the cows to the top half where it says “sum.” The student also has the option of breaking groups and re-grouping. For example, if there is a group of 10, the student can break up the group by pressing the “break group” button so there are 10 individual cows. They may then, for instance, choose to regroup cows by groups of 5 by clicking the “group” button. A space ship comes in and automatically groups or breaks apart the cows. Once all cows are dragged into the sum section, the student presses the “check” button and a box appears that says, “How many cows are in the field?” A calculator is shown and the student clicks the number of cows they think there are and presses the check button. The applet verifies this amount. If the amount is too much, the applet will tell you it’s too much. Likewise, if the amount is not enough, the applet will tell you it’s not enough.  Once the student correctly answers the problem, they are done with the problem and the correct equation is displayed and orally stated. They then click the arrow button for a new addition problem or if they click the wheel button, they can create their own addition problem. In the subtraction category, there are also two halves but the top half is split in half again. A subtraction problem is displayed on the bottom half and the word “difference” is displayed on one of the top halves. For example, the problem 14-9 may be displayed on the bottom half. The number of cows displayed is based on the first number (the highest number) of the subtraction problem. For instance, in the 14-9 example, 14 cows would be displayed on the bottom half. Students first have to select the number of cows that are being subtracted. For the 14-9 example, the student would click on 9 cows. They would then drag those cows into the difference section. Students may find that they need to use the “break group” or “group” buttons. For example, if there are 14 cows and there is a group of 10 and 4 individual cows, the student has to break apart the group of 10 in order to drag 9 cows into the difference section. After the student drags the cows into the difference section, they select those cows and press the “subtract” button. A space ship comes in and automatically moves those cows to the other top half (so the difference section is now empty). The student then selects the remaining cows that are in the bottom half and drags them into the difference section. In the 14-9 example, the student would move the five cows that are shown on the bottom half into the difference section. Then, the student presses the “check” button and a box appears that says, “How many cows are in the field?” A calculator is shown and the student clicks the number of cows they think there are and presses the check button. The applet verifies this amount. If the amount is too much, the applet will tell you it’s too much. Likewise, if the amount is not enough, the applet will tell you it’s not enough.  Once the student correctly answers the problem, they are done with the problem and the correct equation is displayed and orally stated. They then click the arrow button for a new subtraction problem or if they click the wheel button, they can create their own subtraction problem.

             With teacher demonstrations, I think this applet is easy to use and engaging for students. Once students practice with it, they will likely understand what to do in no time! Also, the oral directions and the clear buttons with corresponding pictures (for example, the “select” button says “select” and has a picture showing an arrow) are helpful to make this applet easier for students. The visual presentation of this applet is appealing for students. The bright colored green grass background and brown spotted cows make this applet exciting for students and can contribute to a more positive math learning experience. With a bright colored background and cows, students may be more visually stimulated and drawn to the game (which may help them stay engaged for a longer period of time). While I think this applet is great for young elementary students, as described below, I think the main challenge for students is that there are so many buttons that it can be overwhelming or confusing. If teachers do not model how the applet works and clearly explain what each button is used for, students will likely be lost, irritated, overwhelmed, and lose focus more quickly (however, once teachers do explain it, I think students can successfully work through this applet independently).

Critique:

            I think this applet is a wonderful tool to aid students in their number and operation knowledge. This applet does a great job at helping the students practice and master their counting, adding, and subtracting skills. It is truly interactive by allowing students to select, group, and/or break apart the cows. This applet isn’t a matching game nor is it a simple “circle the correct answer” game. Rather, it requires that students think about how they can group numbers and different ways they can group numbers when adding or subtracting. Plus, this applet is highly engaging because most young students like physically “doing” something rather than watching a video for example. Also, most young students are simply engaged by the cute cows and cool space ship! In addition to the fact that this applet is interactive, engaging, and visually appealing, there are many other strengths of this applet. First, I like how that once the student checks their answer, there is immediate feedback given (it tells the student if their answer is not enough or too much). I especially like how the applet doesn’t tell the student the correct answer if they are incorrect. This forces the student to rethink the problem. I also like how there is no timer so students can take all the time they need to practice the problems they are given. Additionally, there are instructions for teachers to read, which are beneficial for them to understand the applet as well as helpful for them when they explain how to use the applet to their students. Moreover, one aspect of this applet I find particularly interesting is that in the addition and subtraction categories, the student (or teacher) could plug in their own addition or subtraction problem. This would be a great assessment tool for teachers! For instance, a teacher can create a worksheet that has a set of addition problems and subtraction problems. Students will have to plug these problems in and solve them. This would allow the teacher to see if students are understanding the content or if they are struggling. Moreover, I like how this applet gives the instructions visually and orally so students can see and hear them (which helps students who are visual and auditory learners). In addition to the fact that this applet has many buttons and may be confusing for children to follow, one other weakness is that when teachers or students create their own addition or subtraction problem, the highest number they can type in is 20. For students who have mastered the skill and are ready to be more challenged, this can be a downfall (because they are continuing to practice what they already know).  Despite these weaknesses, I think this applet is effective overall for both students and teachers. It can help students develop their basic counting, addition, and subtraction skills and help teachers gain a better understanding as to whether the students are grasping the content. I think this applet is ideal for students who are beginning to learn about addition and subtraction (and grouping and regrouping). As they learn about this concept from their teacher through discussions, worksheets, and activities, this concept can be reinforced through them engaging with this applet.

Applet #2: Base Blocks

Source:

            -From the National Library of Virtual Manipulatives

URL: http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html?from=category_g_1_t_1.html

Summary:

           This applet is intended for grades Pre-K-2 and the math content is number and operations specifically focusing on illustrating using base ten blocks to represent a number. Students are working on representing a number, counting, grouping, and place value. The applet is a grid where the student (or teacher) can adjust the number of decimal places (either 0, 1, 2, 3 decimal places; for example, 9, 0.9, 0.09, or 0,009), the bases (base 2, 3, 4, 5, or 10), and the number of columns (2, 3, or 4 columns; for example a rod and a unit is two columns, a flat, rod, and a unit is three columns, and a cube, flat, rod, and a unit is four columns). Once a student adjusts their settings, they click “show a problem” and the directions appear saying “Use blocks to show: _____” (in the blank, a number appears). Students then have to use the base blocks to appropriately represent the number. There is a “clear” button, which erases their work completely (this may be useful if a student realizes they made many errors), and there is a trash can where students can drag a block they don’t need (for example, if they accidently have 5 units instead of 4, they can drag one unit to the trash can). When students finish a problem, they can click the button that says “next problem.” As a student drags the base blocks, numbers appear on the right side showing how many they have so far. For example, if the problem is “77” as students drag 1 rod out it shows “10,” as students drag another rod out it shows “20,” etc. In short, it shows students how many their base blocks equal as they drag them. Therefore, students know when their answer is correct (immediate feedback) when their total equals the problem total. Also, a student can drag between columns to break down the cube, flat, or rod. For example, if the problem is “39” and the student chooses to drag a rod into the unit column, the rod will break up into individual units (the student will then have to put one unit in the trash for the problem to represent 39).

           This applet is very easy to use. There are no instructions for students or teachers. However, teachers can look at it and easily understand what to do, and students can figure it out by simply exploring for themselves (without any teacher instruction). There are only a few different buttons, and they are easy to use. The visual presentation of this applet is unattractive. The background is all gray and the base blocks are blue. However, it is nice that the blue pops out and that the base blocks are clear and easy-to-see. There are no other colors or stimulating visual features. One main challenge this applet has for students are the various decimal places, bases, and columns to choose from. Students may be overwhelmed or confused when they see decimal places, bases other than base 10, or multiple columns (that show a cube or flat, for example). 

Critique:

            This “base block” applet is a great instrument for students to use as they explore numbers, counting, grouping, and place value. It is great because it is interactive, allowing the students to drag the base blocks to represent the number given. Students are able to really analyze a number. Rather than just being given the number “32” and circling 32 ducks in a pond or making 32 dots for example, students are thinking about what 32 means. It means there are 3 tens and 2 ones, and they can represent it by using 3 rods and 2 units. In addition to the fact that this applet is interactive and allows students to critically think about numbers, there are other strengths as well. As mentioned above, I like how the applet counts with them as they drag the base blocks out. I also like how the applet can be easily adapted for students. For example, a student who is struggling or lower achieving may work with base 10 blocks containing just rods and units and no decimals. A student who is excelling and/or is high achieving or gifted may work with base 10 blocks containing flats, rods, and units (and maybe even cubes) and/or decimals. While I enjoy the various adaptations this applet has, I also see these adjustments as a weakness. Students may be overwhelmed or confused when they see decimal places, bases other than base 10, or multiple columns (that show a cube or flat, for example). Specifically, at this age, I believe students should only be working with base 10 blocks so I do not like that other bases are even an option. I think it can confuse and even hinder student learning. Another weakness of this applet is that it does not allow the student (or teacher) to create a problem. Problems are automatically given. This may be nice if the teacher likes the problems this applet presents, but otherwise, it limits the students to only the applet problems. This limits students’ exploration because they cannot create their own problems to further analyze or discover the content. Also, teachers are limited because they cannot create problems to assess student learning. Assessments may be challenging, because all students will have different numbers and the teacher cannot choose which numbers students will get. I think teachers could assess students by having them record the problem they are given and draw how they represented the problem using base blocks. Or, the teacher can give students problems, have them practice using the applet, and then have them draw their representation using the base blocks. Another weakness of this applet is that there are no written or verbal directions. Teachers can easily figure out how to work through the applet, but students may struggle (so teachers should explain how this applet works before students explore independently). Also, some of the words are hard to read (such as “columns”), and since there aren’t verbal directions, teachers must make sure they review this vocabulary with the students prior to them using the applet. Although I mentioned many weaknesses, I do think this applet can be valuable for students’ to explore numbers, counting, grouping, and place value. Students can delve deeper as to what makes up a number and learn how to represent that number with base blocks. As students progress in math (for example, as they work with addition and subtraction), their knowledge of grouping and place value that they learned working with this applet (along with class activities), will come in handy. I think this applet is ideal for students who have already been introduced to base 10. I don’t think it would be appropriate to have students use this applet if they haven’t been introduced or practiced with base 10 before. Once students have worked with base 10 (and have worked with physical manipulatives), then they can further practice their number knowledge through this applet.

App #1: Teaching Number Lines

Source:

            -From the iTunes store-free app

URL: https://itunes.apple.com/us/app/teaching-number-lines/id492603378?mt=8

Summary:

           

         “Teaching Number Lines” by Aleesha Kondys is a free app that can be accessed from the iTunes store and downloaded to iPhones or iPads. This app is targeted for young students (I would say ideally kindergarten and first graders). Students work with number lines, which is a visual model, to help with basic math skills such as counting, ordering, addition, and subtraction. There are various math activities to choose from in this app: Jump forwards, jump backwards, sequencing skip counting (practice), skip counting (test), and teaching tool. When students click the “jump forward” lily pad, directions appear on a chalkboard saying, “Jump forward to show __________” (in the blank is an addition problem; i.e. 3+9).  A number line (0-15) is displayed and the frog is placed on the first number. The student has to move the frog to the total (ex. if the problem was 3+9, the frog would already be placed on the 3 and the student would have to add 9 and place the frog on 12). When the student is correct, the number turns green, a dinging noise is heard, and the app will show something such as “Bravo!” or “Nice work” (immediate feedback). When students click the “jump backward” lily pad, directions appear on a chalkboard saying, “Jump backwards to show: ________” (in the blank is a subtraction problem; i.e. 13-3). The rest of this activity is very similar to the “jump forward” activity. A number line (0-15) is displayed and the frog is placed on the highest number. The student has to move the frog to the difference (ex. if the problem was 15-3, the frog would already be placed on the 15 and the student would have to take away 3 and place the frog on 12). When the student is correct, the number turns green, a dinging noise is heard, and the app will show something such as “Bravo!” or “Nice work” (immediate feedback). When students click the “sequencing” lily pad, directions appear on a chalkboard saying, “Complete the number line.” A number line (0-15) is displayed with missing numbers. The missing numbers are shown on frogs above the number line. Students have to drag the frog with the number in the correct location on the number line. For instance, the number line may be missing numbers 4, 7, and 2. These numbers would appear on the frogs. The student would have to drag the 4, 7, and 2 in their appropriate spot on the number line. These sequence problems may have 2, 3, or 4 numbers missing (depending on the problem given).  When the student is correct, the number turns green, a dinging noise is heard, and the app will show something such as “Bravo!” or “Nice work” (immediate feedback). When students click the “skip counting (practice)” lily pad, directions appear on a chalkboard saying, “Jump by _____” (either 2, 3, 4, or 5). A number line (0-15) is displayed and a frog is displayed at the zero of the number line. If the chalkboard says “Jump by 3’s” the number line will already have the numbers highlighted by 3’s and the student has to practice moving the frog by 3’s. For example, 3, 6, 9, 12, and 15 would be highlighted in green. The student would then practice moving the frog from 0 to 3, 3 to 6, 6 to 9, 9 to 12, and 12 to 15. On the chalkboard, the student can click the 2, 4, or 5 to practice counting by those groups. This activity is simply practice. When students click the “skip counting (test)” lily pad, directions appear on a chalkboard saying, “Jump by ______” (either 2’s, 3’s, 4’s, or 5’s). A number line (0-15) is displayed and a frog is displayed at a certain number (this number changes). For instance, the frog may be placed on the number 1 or it may be placed on the number 4. If the frog is placed on the number 1 and the chalkboard says “Jump by 2’s,” the student has to move the frog by counting by 2’s (so the student would move from 1 to 3, 3 to 5, 5 to 7, 7 to 9, 9 to 11, 11 to 13, and 13 to 15). As the student moves the frog to the correct number, that number gets highlighted in a green color (so as the student moves the frog from 1 to 3, the 3 turns green). When the student is correct, a dinging noise is heard, and the app will show something such as “Bravo!” or “Nice work” (immediate feedback). The last lily pad is known as the “teaching tool” lily pad. This activity has frogs with a blue dot, yellow dot, and red dot (and as little or as many of these frogs can be used). There is a blue and a red marker (which can actually be clicked on and drawn with) and an eraser. Also, there is a number line (0-15) displayed. The student or teacher can work on any of the math skills shown in the other activities (ex. addition, subtraction, sequencing, and/or skip counting).

              This number lines applet is easy to use. Students can figure it for themselves by clicking the different lily pads and exploring the various activities (without any teacher instruction). Teachers can investigate the app and easily understand what to do in each of the activities (they can also read the directions on the chalkboard). Plus, teachers can go to the iTunes store or Google and research this game to find out more detailed information about the activities. Moreover, there are only six lily pads students can click on, so there are not too many buttons students need to worry about using. The visual presentation of this applet is highly attractive. The home page is bright blue with 6 lime green lily pads. Within each activity (except the teacher tool lily pad), there is a chalkboard that displays the directions and a clear number line below. The frog (s) in each activity are a dark green and cream color, which also add visual interest. With these bright colors and fun features (such as the chalkboard and frogs), this app can help students be more engaged and interested in the math content. One main challenge this app has for students is that there are no oral directions. This is a big problem since most of these students are still learning how to read. Plus, most of the words displayed are not appropriate grade-level words (they are too challenging). Therefore, a teacher or aid must recite the directions aloud, which is an extra hassle and takes away some of the students’ freedom to independently explore and have fun with the app.

Critique:

            This app is awesome for students to use as they work on counting, ordering, addition, and subtraction. Not only is it interactive, engaging, and highly kid-friendly as well as visually appealing, but it also allows students to have a visual representation through a number line. With a wonderful visual model such as a number line, students are likely to visually see the problem, the physical position of a number, and recognize patterns and relationships. Plus, once students really start to work and practice with the number line, often it will likely be a mental image they will remember in their heads and can use almost as second nature. In addition to the fact that this applet is interactive, engaging, visually appealing (with a great visual model-the number line), there are other strengths to be noted. The “skip counting (test)” lily pad is a great way to assess students’ counting skills. A teacher can assess students counting skills by watching them do this test activity or by having them write what the directions in the chalkboard say (ex. “Jump by” 3’s”) and have them draw on a pre-made number line the frog moving by 3’s. Teachers can assess student learning of the other activities in this app the same why (either by observing them on the app or by recording the directions and filling out a pre-made number line). Moreover, one of my favorite activities on this app is the teaching tool. Teachers can give students a counting, addition, or subtraction problem and watch as the students work through this problem on the number line. Or, teachers can work one-on-one with a student or in small groups and practice/review the concepts in the activities. This may be especially helpful for lower achieving and/or struggling students. While this app is appealing in many ways, there are some weaknesses to mention. As previously stated, one huge factor to be aware of is that there are no oral directions to go along with the words. Since these students are still so young (K-2), many of them are still learning and progressing in their reading. Therefore, a teacher must recite the directions aloud. Another weakness I noticed in the “jump forwards” and “jump backwards” is that the students may just drag the frog where it goes but not actually understand the content. For example, in the “jump forward” lily pad, students may just drag the frog until they hover over the number that turns green (green means they got the correct answer). If students were to do this, they are not gaining any mathematical knowledge. Hopefully, students will actually think through the problem and use the number line appropriately. Also, teachers can monitor as well as assess students to make sure they are engaged and understanding the content. One other weakness of this app is that the numbers on the number line do not go very high (the number line only goes to 15). This is great as students are beginning to practice counting, ordering, addition, and subtraction. However, once students start mastering these skills, more challenging problems that are beyond the scope of this app will need to be provided. Even though I stated a few main weaknesses, this app can be advantageous for students as they are working on basic counting, ordering/sequencing, addition, and subtraction. The number line is one important strategy/model students can use to help them understand numbers and operations. If students continually use models, such as number lines, to help them with numbers and operations, they will probably have a better understanding for the problems and begin to do these problems mentally. I think this app is ideal for students who are just being introduced the concept of counting, ordering, addition, and subtraction. As students learn about this concept from their teacher through discussions, worksheets, and activities, this concept can be reinforced through them engaging with this app.

Final Thoughts:

          While each of these applets and app have their strengths and weaknesses, I could envision myself using all of these in my future elementary classroom to aid, positively contribute, and reinforce math concepts to my students. I want my students to realize that learning math does not need to be boring or a complex process. Rather, exploring math concepts online through applets and apps will show students that math can be fun and they will have a richer, more interactive, and more meaningful math learning experience. Plus, nowadays students thoroughly enjoy when they get to use technology, so by having students use these math applets and apps, they will be excited about their learning. I hope to use these applets and app in my future classroom amongst many others that I learn or find out about along the way in my teaching journey.

Article Discussions

Article Discussion #1
A Model for Understanding Understanding in Mathematics by Edward J. Davis

            This article is about understanding the mathematical process through different kinds of “moves” in teaching mathematics. It is essential that teachers gain knowledge about how students are understanding so it can help guide them for future planning of instruction and assessment of learning. The word “understanding” is a broad term and has various definitions (for example, it may mean being able to state it in your own words or giving examples of it). Also, understanding is a range; students may have full understanding, no understanding, or they could have only a partial understanding. The “moves” that a teacher uses is essentially how the teacher explains the mathematical information to the students. In school mathematics, typically concepts, generalizations, procedures, and numerical facts are the mathematical knowledge taught. Within each of these areas, the article presented two levels. Level 1 was where students are grasping the basic knowledge, such as the “what” (ex. understand “what” the generalization says) whereas Level 2 digs a little deeper and may discuss the “why” (ex. understanding “why” the generalization is true) or “characteristics”  (ex. understanding characteristics of the concept). Also, I like to think of these levels as building blocks. It’s a step-by-step process where students’ understanding gets progressively deeper. For example, when teaching “concepts” (level 1: examples and nonexamples of the concept), students understand a concept when they can: give or identify examples of the concept, defend choices of examples of the concept, give or identify nonexamples of the concept, and defend choices of nonexamples of the concept. You can see that the understanding starts basic and gets more challenging. However, it is important to note that students’ understanding may jump around (they may understand one step or some aspects of one level before understanding another; i.e. they may understand some aspects from Level 2 before Level 1). The article also discussed how elementary teachers’ instruction starts at physical then goes to pictorial and then moves towards symbolic representation. I appreciated how the article then went on to say that when students are at the symbolic level, they should not stop. When they are at this level, they should be able to explain and interpret their work (through pictures, number lines, counting objects, etc.).

            This article was insightful for me as a future elementary teacher because it is critical to know how students are understanding the content. Through the various “moves”, teachers can better determine where certain students are getting stuck in their learning. When teachers understand the moves at which certain students are getting stuck, a few things happen: 1. A teacher can better prepare for future instruction and assessments, 2. A teacher can better diagnose a student’s understanding of a mathematical concept, generalization, procedure, or numerical fact, and moves can help show the student’s knowledge, and 3. A teacher can pick moves and instruct aids or tutors to focus on those specific moves with certain students. I was captivated by how the word “understanding” is so detailed. I learned that there are so many levels to understanding and students will likely be at various levels of understanding, which is important to keep in mind when teaching in order to properly meet the needs of all students. In my future classroom, I think utilizing “moves” will be beneficial so I can gauge where my students strengths and weaknesses are and where I need to reteach certain moves.

Discussion questions:

1.     While this article showed the importance of using “moves” to understand students’ mathematical knowledge/comprehension, what challenges may moves present in the classroom?

2.     If “moves” seem to be successful in mathematics, would a similar process be successful in other subject areas (i.e. language arts, science, etc). Why or why not?

References:

Davis, E.J. (2006). A model for understanding understanding in mathematics. Mathematics Teaching in the Middle School. 12(4). 190-197.

Article Discussion #2
Thinking Through a Lesson: Successfully Implementing High-Level Tasks by Margaret Schwan Smith, Victoria Bill, and Elizabeth K. Hughes

            When teaching mathematics there is an emphasis on providing students with rich, critical thinking tasks where they are given opportunities to use reasoning and problem solving skills. However, when these thought-provoking, often challenging tasks are implemented in the classroom, research has shown that much thinking and reasoning is lost and no meaning is made. Creating and implementing these rich and challenging tasks are often difficult but they are important for students’ mathematical growth. That is where the TTLP, the Thinking Through a Lesson Protocol, comes in. TTLP is a process to help teachers teach high-level tasks and develop a detailed lesson plan prior to teaching. There are three aspects to the TTLP: 1.  Selecting and setting up a mathematical task (teacher identifies the math goals for the lesson and thinks about how students will work through the problem), 2. Exploration of the task (students work individually or in small groups and are asked questions to assess their current knowledge and to guide their learning), and 3. Sharing and discussing the task (class has a whole-group discussion of the different solution strategies they used). While the TTLP may be time-consuming, it is beneficial and effective. For Part 1, selecting and setting up a mathematical task, the first thing a teacher should think about is: What are the mathematical goals for the lesson? For Part 2, exploration of the task, the teacher should create good questions to ask during instruction to help students focus on the main mathematical ideas that are a part of the task they are exploring. The questions created should assess what students understand about the problem and advance them towards the objectives/goals of the math lesson. For Part 3, sharing and discussing of the task, it is important for the teacher to think about the various ways the task can be solved. The teacher should brainstorm the correct and incorrect strategies a student uses to solve the problem. TTLP helps teachers think about what the students are doing and aids teachers in guiding students to make sense of the math content as well as advance their math understanding.

            After reading this article, I learned that while TTLP may be a long and time-consuming process, it is worthwhile. The more teachers use TTLP, the more likely they can start internalizing this framework and plan effective lessons. Furthermore, I learned that anticipating student responses/questions ahead of time and preparing questions prior to the lesson to ask the students is crucial. These prepared questions will help guide the students with the problem at hand while still allowing them to explore the answer for themselves. I can envision myself using the TTLP process in the future because it will allow me to think about how my students may think about the content. It will also help me be more prepared to ask them appropriate questions which still allows them to explore while also guiding them towards the math objective of the task. I believe that with the TTLP process, I will be more prepared and the lessons I teach will be more successful.  

Discussion questions:

1.     This article explains the importance of teachers brainstorming possible correct and incorrect student responses. However, how would you handle a situation where a student’s response is not one you prepared for?

2.     In Part 3, sharing and discussing the task, how will you, as the teacher, lead this class discussion? What do you think is the most effective way? (i.e. Do you think you should have each small group present their strategies? Would you have one student representative from each small group share their strategy? Would you, as the teacher, present strategies and ask if any of the groups did any of the strategies you presented? Would you have students debate which strategy they think is the most accurate?)

References:

Smith, M. S., Bill, V. Hughes, E.K. (n.d.) Thinking through a lesson: Successfully  implementing high-level tasks. Designing and Enacting Rich Instructional Experiences. p. 11-18