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.