After asking various people measurement questions with a short assessment that I designed, I have developed a better understanding of how students and people around us think about measurement. The questions that I developed on the assessment were a compilation of ideas from my original lesson learning objectives. I wanted students to learn both systems of measurement, the types of units involved in each system of measurement, and what types of tools to accurately use for each unit when measuring. While this information is important to know, adding in estimation tactics, and conceptual thought of how measurement is used in the world around us is also important. This formative assessment consists of questions that cater to these big ideas, and ones that allow the developer to see a further understanding of where lessons may need to begin and end within the content. While many responses vary, the range of people that I gave the assessment to was highly differentiated. Some people were much older, and others still in high school. These responses have allowed me to determine many new outlooks on measurement, and classify different responses with categories.   

One of the big ideas is that there are always connections to be made to real life when learning about measurement. There were many references to building and cooking within the responses. Also, most responses that related real-life experiences to measurement didn’t have the strongest knowledge of units, difference between systems, and estimating tactics that measurement develops. However, their relationship to real-life, allows me to understand that they are confident in what they are measuring and how to measure, to be able to use it and apply it to other areas of life. Other responses either lacked connections to real life, or proper content. While most of the responses that people gave were correct, there was high variation of accuracy that people made as to why we even measure. These responses are ones that as a teacher should be re-looked. These students don’t understand the main motive to measuring. If that is not apparent why even learn the content involved.

The following are specific associations and further descriptions of each group that was classified. Misconceptions, class relations, questions, and further thinking are all explained.

 

Able to Confirm Measurement Reasoning- 4 students

                  The highest or most advanced group shows a good understanding of the content presented can relate and use this content in their lives. They were placed in the category Able to Confirm Measurement Reasoning because they can look at different measurement aspects and confirm what is presented, accurately answer the question, and determine where they have seen and used this information before. These students are ready to learn more about measurement, explore more tools used to measure, and accurately confirm most measurement content. These students answered all components of each question asked. Their thinking was clearly presented, and shown in a way that is easily readable. They can use different tools and accurately measure. These people need to further advance in different types of measurement, and also new measurement tools. They are ready to reconfirm ideas and move onto new material. A lesson that structures around what they know, and how to apply it to real life is a good place to start with these students. As a teacher I would learn more about what else they know and don’t know but also activate prior knowledge.

Some misconceptions that may be included would be how to estimate accurately and in the most efficient way. Some connections that were made to estimation were only developed because the exact mileage of their road trip was known. For example, when I asked them to estimate how far that they traveled during their road trip and how do you know its accurate to the exact distance, one response that I got was, ’90 miles, yes it is accurate because I started at mile 150 and I exit at mile 63.’ Since this person was too aware of their distance traveled, they knew that their estimate was correct. In this case, was it even an estimate? They may not understand how to think about estimation- but when most measurement content is known, it is usually clear to these people where about their estimate should be.

These students also need to be presented with a high mathematical ceiling. As we talked about this in class, students are engaged the entire time of the lesson and relate certain examples to other experiences they have had. While a low mathematical floor aspect to instruction will keep these learners confirming what they already know. These students are recognized at the symbolic level of representation and able to accurately connect content and other areas of their lives and math. These students are in the progression of knowing it well and understanding all terms. Teachers need to be there to push these students in the classroom to extend their learning and later help them further explore topics presented.

 

Questions to ask these students:

• What connections do you notice between various forms of measurement?

• Where do you use measurement outside of the classroom? How?

• Estimation is a powerful tool in the real world. How could you use a measuring tool to estimate the length of various items?

• Draw a diagram that shows what measurement means to you?

• Have you ever heard of an odometer? What is this tool used to measure?

  • While these questions may be tricky- students are thinking about what they know and applying it elsewhere.

 

Real Life Connectors- 6 people

                  The second group that I categorized was labeled Real Life Connectors. These responses that I received didn’t have the strongest connections to measurement content but they were able to remember where they measure outside the classroom. There were some people within this group that had a better understanding of content than others, but their responses were among the same types of answers. All components of the questions were not fully answered, and questions that should be understood, if they confirm outside measurement relations, were not accurate. This tells me that they may just need some reiteration of content about measurement. While their schooling may show influence them to relate measurement to real-life more than see the relevance in the content, other students were more condiment in the content rather than applying the knowledge elsewhere. There was a clear balance between both situations within this group.

                  Some examples of these student’s answer to the question, what did you use these tools [the ones previously listed] to measure? - ‘Included a 2 by 4’, ‘cooking at the restaurant I work at’, ‘measuring the distance on my archery target,’ or ‘finding the thousandths of an inch’. As one can notice the variation of responses, most of them were related to when they had to measure something outside the classroom. That was the situation they were thinking about when asked about measuring tools. Also, the measuring tools that were listed were ones definitely taught in the classroom, but never fully explored. Examples like, measuring cup, or odometer.

 

Misconceptions within this category could comprise of not knowing all units of both measuring systems on a ruler, or having two uniform systems of measurements makes things easier. Does it make things easier with two forms of measurement? What is the point to having our own U.S. customary system of measurement?

 

Questions:

• Form an opinion as to why two systems of measurement are helpful, or not helpful to have.

• Where was the last time that you used a ruler? What was it for, and what units did you use when measuring?

• What is the purpose of estimating? Why do you not want to know the exact measurement before you estimate?

• What other measuring tools do you use outside of the classroom.

 

When discussing the likelihood line towards the end of our measurement unit in class, I was able to clearly see my mistakes that I first made and how I better understand what I was originally thinking. Within my class, I would give students a likelihood line before and after our measurement unit. This would help them do the same thing that it did with me- readdressing ideas that I previously was confident in. I feel that people within this category are able to think conceptually about measurement, but they need to rethink measurement aspects. Following through a likelihood line with measurement qualities could allow them to build on their foundational thoughts of measurement and further develop their findings throughout the unit.

 

Developing Measurement Understanding- 5 people       

Finally the third group that I categorized is known as Developing Measurement Understanding group. This group needs to build both content knowledge and real-life connections within measurement. If they understand how this content works together, they can later connect it to measurement outside of the classroom. These relationships develop student’s further progressions in reading certain measurement-based questions and make it able to think clearer about them.

                  These students were able to think conceptually about measurement, but their performance on the assessment was very poor. Certain questions were not fully answered; no thought was put into answering the questions, and various aspects of measurement content were not clear. For example, a response to ‘Where was your road trip?’ and the response was GPS, this shows that either the questions was unclear, or that their laziness took over and had them not answer the question properly. They need to further develop different parts of measuring tools, measurement units, and estimation skills, and also read directions carefully in later assessments.

                  Within the learning progressions students fall under the enactive component of levels of representation- process learning progression. This is where most students can work with manipulatives and act out what they are solving and doing. This group would definitely be able to do this if I gave them many different measuring tools, and items to measure. However, will they use the correct measurement, unit, or system when asked to do so? Another response given when asked where did you use these tools to measure, someone responded with, ‘geometry’. This was one of the responses closed to real life situations within this group. They related it within math, but never fully developed their thought on how they related measurement to geometry. These aspects would need to be cleared up and gone over before students would later develop questions.

                  Misconceptions among this group are common. They may think there is only one system of measurement that we use today. Answers including ‘there are none’ in reference to differences and similarities between U.S. customary and the metric system. These simple yet big misconceptions could form other components of their thinking about measurement. As a teacher I would start with the simplest aspects of measurement and work more abstract.

 

Questions:

• Where have you used measurement outside the classroom?

• What are the two systems of measurement?

• What are the most popular tools used to measure?

• Have you every learned how to estimate? What is estimation?

While some of these are similar to the ones from my formative assessment, they may need to rethink these questions, really relate to them, and then piece together various measurement ideas. These are good areas to begin with a group like this.