Article Review: The answer is 20 Cookies. What is the question?

In the article, The answer is 20 cookies. What is the question?, a new approach to problem solving was presented to teachers from different classrooms. They were then given the assignment to try it in their classrooms to see what would happen. This approach uses the working backwards strategy where the answer is given and you have to determine ways of getting the answer by creating the problem. The teachers used this technique with their students and stood amazed. For instance, one student was given the answer 245. That problem that the student constructed was, “There are 245 red cars are in a parking lot. 200 blow up how didn’t blow up? (Barlow and Cates, 2007)

The teachers had a carefully selected math problem that would yield critical mathematical reasoning and the use of prior knowledge. Because the teachers had selected a problem that would have a wide variety of solutions, it acted as a prompt to get the students to think creatively and to think of new ways to incorporate information they already knew. They were enabled as learners to take existing knowledge and come up with a problem that was innovative on their own. (Barlow and Cates, 2007) This is supported by the standard in that teachers are encouraged to promote problem solving by compiling innovative information and prior knowledge. They do this by selecting math problems carefully that will help their students think outside the box.

Some of the students when given the answer 20 cookies wrote down “How many cookies?” Others would give the answer 10+10. (Barlow and Cates, 2007) Although they were giving an answer, they needed more explanation in order to determine the appropriate solution to the problem. In the article, teachers found that sometimes there is a need for further instruction and guidance and the students can alter problem solving strategies and methods. They could then relate to the answer given to come up with a creative and well thought out problem. The problem solving standard supports this principle of being flexible to changing strategies because different problems will need different strategies and the students need to be able to know how to differentiate between the strategies in that way. The students also need to be open to reconfiguration in their problems because there could be a better way of solving.

In the article, the students had to come up with the problem, as the answer was already given. They were encouraged to be creative in their thinking in order to come up with a problem that would be effective. Their problems were about real life situations which drew correlations between math and real life which caused them to think of how to problem solve in other situations. (Barlow and Cates, 2007) The problem solving standard reinforces this idea in that students should be able to come up with valuable solutions to problems that they will face not only in math, but in real life situations.



Barlow, A. T. and Cates, J. M. (2007). The answer is 20 cookies. What is the question? Teaching Children Mathematics 13(5), 252-255.

Problem Solving

Problem Solving is used anytime the means for attaining an appropriate solution is unknown. Problem Solving enables students to glean from prior knowledge, and wrestle with problems to find the correct conclusions. The great thing about problem solving is that it helps students understand more mathematical ideas and concepts, but it also helps students grasp ways of processing to problem solve in real life situations. Problem solving is a huge part in math and it should penetrate every area.

Compile Innovative Information through Problem Solving

It is imperative that teachers choose wisely the problems they present their learners. These problems should be worthwhile and be relevant to their lives in order to introduce new mathematical concepts. Students can engage in math that is close to home in that what they are learning is useful to them. Students will grasp new concepts with less difficulty if they can draw correlations to their own experience.

Create effective solutions to problems that come up in math and other situations

Those who excell at problem solving tend to bring situations under scrutiny and there pose problems based on their observations. Assumptions can be made by looking at a problem with little investigation, but the further one delves into the problem the more accurate the outcome will be. Educators have the responsibility of cultivating an environment within the classroom in which students are free and encouraged to create problems that align with their own lives to solve. They should be exhorted to investigate, discover new insights, try new things, and take risks in the classroom.

Relate and adjust many effective problem solving strategies and methods

There are many different problem solving strategies which include: working backwards, guess and check, the use of diagrams and patterns, etc. Instructors should be teaching these strategies to their students along with other problem solving methods. They should assign tasks in class that use a variety of strategies so that students can compare different ways to solve a problem and thereby the students become fluent in using them by the time they reach the higher grades. Different types of problems are going to require different types of problem solving methods. Classroom discussion about problem solving is going to make this a norm in the classroom and help students to be able to internalize what they have practiced and observed.

Observe the process of problem solving and be able to draw personal insight and application

Effective problem solvers are continually engaged in observing their actions and processes, making changes that they see that are necessary, read and reread, and ask questions. They also keep track of their progress and are open to change because there may be a better way of doing things. When the teacher asks probing questions to make sure his or her students understand concepts, and that the learners have gone into much depth in their thinking, they create an atmosphere where students feel free to assess their own work and thinking. It is very important that learners lay hold of reflective practice. By doing this, they will become more efficient and effective at problem solving not just in math, but in their own lives.

The Variable Machine

The activities in these video clips serve the purpose of showing that the values of the variables have an effect. In the activity, the students can change the variables associated with the given values in order to increase or decrease. The activity is also a hands on learning tool which causes the students to work with problem solving both individually as well as collaboratively. This also enables the students to grapple with mathematical problems, ideas, and solutions; giving them a foundational understanding of math as it connects as a whole.

Discourse #1
How effectively does the teacher use questioning to help students develop mathematical understanding?
The teacher does an excellent job of asking students questions that do not yield "yes" and "no" answers. She asks questions that guide the learners without giving answers. When the students are stuck, but on the right track to understanding, she prompts her students with guiding questions that helps them figure things out on their own. I love how the teacher answers the students' questions by asking them questions that help them find the answer they were trying to get.

Discourse #2
What techniques and strategies are used to orchestrate and promote student discourse, and how effective are these strategies?
The teacher gives brief and concise instruction for the students on what they will be doing and then lets the students get started working in groups. As the students are working she walks around the room which keeps the students on task and asks questions to groups as well as to the classroom as a whole. The teacher also listens to the students' solutions and explanations to ensure their understanding. I think that these methods are highly effective in student discourse. The teacher asked strategic questions that caused the students to expand their horizons and think creatively.

Student Learning #1
What strategies were used to assess student understanding?
The teacher used an informal approach to assessment because she didn't feel that it would be effective to have a formal assessment. The educator could walk around to each group and listen to different discussions to see if the students understood. As the students would work in their groups, different questions and ideas would arise which would give the teacher insight and understanding to where the student's knowledge was. The teacher would also have students share what they got for their solutions and how they arrived at their conclusions which would also act as a means of assessment.

I thoroughly enjoyed watching the video and thought that the activity was very creative. I felt like the activity was productive and worthwhile. As I was watching the videos I kept thinking that I never learned math in this manner. We never reasoned with other people or with the whole class how we reached our conclusions. I think if I had learned this way, I would have enjoyed math a lot more. I was encouraged to help students go deeper in their understanding of math and broaden their thinking. I appreciated being able to think critically about the videos in correlation to the questions that were being asked so that I knew what I was looking for. This was a very helpful task.

Article Review

In the article, Support preservice teachers' reasoning and justification, they give examples of classroom settings, and students working through problems. Throughout these examples students are instructed that they will be using prior knowledge and build off of that in order to complete the problems. The teachers are reinforcing within their classroom content that has been presented and expounding on it in order for students to make connections. In the teaching principle, one of the goals of an effective teacher is to help the students be able to correlate previous knowledge to current content that they are learning so that they can see the overall picture of math integrated.
The students are also collaborating in their work to draw from each learner's knowledge and to contribute. This idea was presented in the teaching principle because students should not just be doing busy work, but working together to do meaningful and worthwhile activities. When students work together they learn more and communicate more effectively a solution to a problem.(p218)
In the classroom segment C, the teacher acted as a facilitator. She waited for repsonses, asked questions that triggered more thoughtful explanations, etc. This is an idea presented in the principle because an effective teacher asks thought provoking questions that will spur students on to think more critically.
According to the article, students are to take action to direct their learning in that they ask questions, explain mathematical thinking, understand sources of mathematical ideas, and are responsibile for their own learning.
The article says that teachers should have a deep understanding of the functions of math and the relationship between numbers, operations, functions and how they relate to another. The principle states the same, that the teacher should have a fluent understanding of math as a whole and how it is intertwined.
The article is aligned well with the teaching principle. It was interesting to read this article and see exactly how the teaching principle was applied in these classroom situations with the given problems.

Rathouz, M. M. (2009). Support preservice teachers’ reasoning and justification. Teaching
children mathematics 16(4), 214-221.

The Teaching Principle

Upon reading the Teaching Principle, I learned that in order to be an effective teacher in the area of mathematics, one must possess a strong knowledge base and glean from it to approach teaching while being versatile. The teacher must view every student as a person and a learner; therefore, since not everyone learns the same, a variety of techniques need to be implemented when teaching to ensure quality education.
The teacher must have two specific types of mathematical knowledge. The first type is knowledge about the whole of mathematics. Instructors should understand overall themes, ideas that are conducive to appropriate grade levels, and goals and standards. This type of knowledge lends itself to aid in curricular decisions, helping students understand better, and connect math concepts together. The second type of mathematical knowledge is Pedagogical knowledge. This helps instructors grasp how the student learns math. Teachers should be fluent in a variety of teaching strategies. This also helps in knowing how to approach classroom management and also organizational skills within the classroom.
I also learned that teachers must have a firm understanding on how mathematical ideas relate to one another and to be familiar with the common roadblocks children have with their understanding of math in order to bring insight that will cause students to draw the accurate conclusions.
Teachers hold the responsibility for ensuring their students an environment in which to learn math. They are responsible through their own actions, conversations, behavior, etc. how their students will then think about math. Teachers should be facilitating conversations that will lead the students to think for themselves in drawing conclusions about math, problem solving, and thinking critically about math. Educators should also be choosing activities that are productive for the student to learn and put their knowledge in to practice. The activities should also include real life situations so that the students can deepen their understanding of the content.
Being an effective teacher goes beyond the practical implications to understanding students, listening to them, and observation. When teachers engage in reflective practice they are evaluating themselves as teachers and the classroom to know how to better their teaching strategies and bring more creativity that will be more conducive to the learner. Reflection is vital in the classroom to create the best learning environment possible.

The Assessment Priniciple/Article: On-the-Spot Assessments

The Assessment Priniciple highlights important points concerning the major part of giving assessments to students of all ages, such as improving students learning. It's important for teachers to understand that assessments do not dictate how smart students are, but it judges their performance on that particular subject matter. The Assessment Principle briefly describes how teachers should be careful when deciding which assessment to administer to students. Since every student learns in different ways, teachers should make assessments a normal routine as part of the classroom instruction, instead of a distraction to the classroom environment. As expressed in the reading, assessments should develop students learning and is an important tool for teachers to use when making informative decisions.

I think this article describes how students use their estimation, prediction, identification, observation and number thinking skills. Although this article strongly informs readers about the five main points listed above, it strongly supports how students observe and predict. In the activity explained in the article, the teacher began by recognizing that her students had observation problems which became another problem for them to estimate quantity. Four canisters labeled A-D were filled with differently sized candies. Students were told to predict how many candies were filled in each canister. At the end of the assessment, the teacher suggested when they are told to predict how many candies are in a jar, they should ask the question, "How big are the candies?" Asking a question that is relevant to the study is a change in how math can be taught and assessed.