The link above is an article by Lynne McClure on developing number fluency. This article gives the “what, why, and how” for developing mathematical fluency in students.
McClure writes about how schools are interpreting standards in a way where teachers are feeding students “a pretty rigid and boring mathematical diet”. Formal mathematical algorithms are practiced in pursuit of mastery, but giving these calculations meaning has been put on the back burner.
McClure describes number fluency three modes of criteria: efficiency, accuracy, and flexibility. These criteria lend to effective communication of mathematical concept knowledge.
In one of my classes this semester, we created a Project Based Learning unit called City Zoning. In this assignment, our group created a project where students would analyze economic and engineering data to create a plan for developing a four block area. Students are required to calculate things such as area of buildings, interest on loans, net income, profitability per square feet, how long it would take businesses to pay off start-up loans, and tax income for the city for sustainability. Students will be required to write explanations for decisions made, give a verbal presentation of their plan, and construct a model to scale of the area. By doing this, students will practice efficiency, accuracy, and flexibility of mathematical knowledge.
The difficulty of creating opportunities for mathematical fluency is multifold. Creation of these types of units is very time consuming and finding activities other teachers that address these goals is not easy. There are a lot of activities online, but units such as these are that scaffold math concepts and put them together for solving real world problems are not very plentiful for teachers to share.
As educators, we want to prepare students with maximum readiness for entering the real world. Math fluency is an important part of critical thinking and decision making because math is what I call “the language of logic”. It is a mode of communication that students need to be able to thrive in an ever-changing society. The way teachers facilitate the learning of math fluency can be the difference in improving collective math achievement.
I had the opportunity to read “Making Room for Inspecting Mistakes” (link: https://drive.google.com/file/d/1PjEE0Y8SqMDVIVrETxgLDWCpX00aeDBO/view?usp=sharing) in this month’s issue of Mathematics Teacher from NCTM. The article discusses using mistakes to help students learn. One of the examples it utilizes is choosing a homework problem that is incorrect to go over for the entire class. Choosing these problems is artful because the teacher must ensure that the problem is going to be useful to the greatest amount of people in the class. There may be people in the class that would make the same mistake, others that get a better understanding of how to complete the problem because they did not know how to originally, and even others that understand where the mistake came from and how to combat it. There are three different contexts for leveraging mistakes that the article discusses: review of homework, during a task example, and during exam preparation. In each of these context, mistakes can be capitalized upon to help students grow in their understanding of the content that is being taught. During review for exams, it is a good check to ensure that students do understand what they have learned throughout the unit/semester/year.
This article prompted me to think about how making mistakes can be useful in the mathematics classroom. We have previously discussed how mistakes can be utilized to help students. Yesterday, and earlier today I had the opportunity to attend the SDEA Student Conference in Mitchell. One of the two breakout sessions utilized Breakout EDU. The concept of a Breakout EDU is similar to an escape room, but students are trying to break into a box. They can be bought online for different content areas. However, they do cost $125 so many teachers write grants to get Breakout boxes. Although escape rooms may just be a fad, Breakout boxes can benefit the classroom. After the activity, I began to think about when I would use Breakout EDU in my own classroom. I believe that these boxes could be useful at the beginning of the year to set a standard for collaboration between students, productive struggle, and making mistakes. Furthermore, during this time at the beginning of the year, a box could be useful as a review from the previous year’s material for the students. We saw in the lesson study that students were reluctant to productively struggle, and using a Breakout box could allow the students to start the year off participating in an activity that calls for productive struggle. Additionally, in the theme of making mistakes in the mathematics classroom, students are bound to make mistakes in their search for the answers to the clues. Using an activity such as Breakout EDU would allow the students to understand that making mistakes is beneficial, especially if they persevere in opening the box. Setting a standard for the benefits listed above of Breakout EDU in the classroom would help establish a particular environment in the classroom for the rest of the school year. This environment is aided in being established because after students complete the Breakout EDU, they discuss what went well for them, what problems they encountered, what did not go well, etc. The reflection is what cements the environment. Overall, there are clear benefits to making and going over mistakes in the mathematics classroom, and Breakout EDU could be used at the beginning of the year to establish an environment that promotes productive struggle, making mistakes, and collaboration between students.
Looking through articles on NCTM’s website, I cam across this really interesting one about how a math teacher was questioning his seven year old daughter about division. During that time he asked her what 18 minus 3 was, and while his daughter was thinking his five year old son spoke up quickly and said 15. Although this is the right answer and the young boy deserves recognition for coming up with the answer so quickly, it set something else off in the little girl.
It caused her to proclaim that she was no good at math when the reality is that she is rather good at math. The difference between the boy and girl in this example is that the boy had a natural intuition toward the answer, whereas the girl had to go through a thought process to get to it. This is often time the case on many levels of mathematics. Males often have a natural intuition to mathematics and are easy to quickly get the right answer, but females are often more likely to say something more insightful about the mathematics being performed. I have seen this happen many times from elementary school to college.
One jarring food for thought the article author gave was to have every educator reading the article to think of the best math students they have ever had. There is probably both men and women in that list. The men are probably the ones who naturally got answers, were laid back about homework and tests, and generally just floated by in the class. However, the women were probably the studious, hard workers, and generally cared about how they were doing in the class. It is shocking the bias that there is between men and women!
The author also observed that women often performed better on tests if it were emphasized that it was not as important to have the right answer as it was to have the right thought process laid out. Often times I have wished that professors would grade based on effort put into the class and not just whether I could recreate a proof on an exam.
As much as I would love to spite this article as total bologna and that women learn and approach math the same way as men, it has some interesting and good points. I remember always being naturally good at math, but not in the sense that I could blurt out answers. It was more that I can follow elaborate patterns and draw conclusions from large sets of data. Maybe men and women do approach math differently?
I always said one of the big reasons that I wanted to become a math teacher was I wanted to inspire young women to like mathematics. That goal has not changed. I want to take the information from this article and use it in the sense to emphasize the fact that answers are just as important as processes. Women are more than capable of doing the same mathematics as men, and I want to inspire my students to see that.
As a student, it was never fun to have the teacher specifically call you out for a mistake you made that is for sure, but was it helpful to the whole class? Yes! Especially depending on the specific mistake that is brought to the attention of the class. We have talked a lot about thinking about the mistakes students will make in a lesson before the lesson so that we know how to properly answer the question. We have also talked about expecting certain mistakes and being ready to exploit and explain them to the class. This is a crucial part to student learning as they can further understand the problem and why what they did is wrong. Putting a red X on their homework assignment is less effective than bringing the attention to the class as they are not as engaged.
The saying still stands true that ‘if you had the question, someone else probably had the same question and was too embarrassed to ask’ and so I think the same stands true for ‘if you made a certain mistake, some one else probably made the exact same mistake’ or at least had the same thought process. As long as you, as a teacher, make it known that your classroom is a classroom for mistakes and growing from these mistakes and you do it often enough, your students should begin to not get embarrassed by this and be able to take constructive criticism well: both are very good and rare traits in our society today.
So why is bringing attention to students’ mistakes so important in the classroom you may ask? An article from NCTM states that, “increasing evidence shows that making mistakes creates productive pathways for learning new ideas and building new concepts (Boaler 2016; Borasi 1996).” Students are able to think a problem through, make the mistake and go back and understand exactly where they made the mistake, how they made the mistake, and how to fix the mistake for next time. This critical thinking leads students to a deeper understanding of the knowledge and instead of walking around to every students’ desk and explaining their mistakes, it saves time and shows an acceptance of failure to be able to call the class’ attention tot he mistake and to work through it as a class. Collaboration is a very important skill to teach and for these students’ to have going into their real lives.
The article discusses inspection worthy mistakes which are the mistakes you predict will come up and you know will help lead the class into a deeper understanding. It is also very possible that students will think of new ways to make different mistakes that you did not previously think of, but will provide an incredible learning process through investigation. These mistakes are typically big-idea mistake not computational errors. For example, a student is multiplying matrices and they do it in the same fashion that they added matrices. This is a good time to bring the class together and explain that that is a common mistake, but that is not how multiplying works. This shows that you did not teach multiplying matrices well enough where they could have made the distinction between the two procedures.
Bringing mistakes to the attention of the class helps students collaborate, create a deeper understanding, fix their mistakes for the next time and help students’ take constructive criticism well.
Article found at: https://www.nctm.org/Publications/MathematicsTeacher/2018/Vol111/Issue6/Making-Room-for-Inspecting-Mistakes/
With my group doing our lesson on systems of equations, I decided to do this blog on exactly that. The first article that I found on the NCTM website about teaching systems of equations is titled “Rethinking the Teaching of Systems of Equations.” That article gave me this concept map for linear equations. This map really breaks it down well, splitting it up into representing, solving, interpreting, and conceptualizing. After splitting it up, it goes into further depth of each of those categories. For example, for representing systems of linear equations, it tells you that you can use technology, context, graphing, equations, and table of values.
After the concept map, it gives examples of different uses and representations of equations in figures 2-7. For example, in figure 5, it gives a representation of how you could use them in a consumer context and in figure 6, it gives an example of a graph representation.
The next article is titled “Problems before Procedures: Systems of Equations.” One thing I found interesting is a figure that gives different questions that teachers can use. I like this table because it gives different questions for two different approaches, graphing and algebraic. I believe that letting students struggle in math will help them in the long run. Instead of just giving them the answer when they are struggling, let them work through it. If at that point, they continue to not get where you want them to be, offer a leading question that could point them in the right direction. At no point do I think they should just be given the answer. I think that these questions are good examples of some that you can use.
This is a link to a site from Grand Canyon University. It is a set of guidelines for providing effective feedback to students individually or collectively. Feedback is a very important part to promoting growth, especially in the math classroom. I have heard the following statements numerous times in classrooms:
“Is this right?”
“What is the answer?”
“Did I do this right?”
Students, in a lot of cases, do not want to make mistakes. Finding a way to give students courage to go out on a limb and try things is something all teachers should strive for. A critical element to getting students to take chances is to provide effective feedback to students while they are in the process of learning.
The type of feedback a teacher gives is very important. Timing, amount, and audience should be considered when giving feedback to students. Punctual feedback should be given for things students turn in so that it is not forgotten. It’s important for the student to have the teacher to bounce struggles and ideas off of, but it is just as important for the student to become independent in learning mathematics. If the majority of a class is struggling with the same concept, it doesn’t make sense to address each student individually about the misconception.
From my experiences, it is very important that the student initiates the feedback from the teacher in some way. If the teacher does not know what the student is understanding correctly and where the misconception is, it’s very difficult to provide effective feedback. Students need to communicate what they know to a teacher for this to occur. Often times, however, students do not know what they are missing out on, making this very challenging. This is why the teacher needs to have a positive relationship with the student, understand how to ask questions to get the student to communicate their thought process, and be able to communicate in a way the student can understand.
I think it is important to actually schedule in periodic sessions where students and teacher can talk about thought processes. This gives students and the teacher an opportunity to, if nothing else, build the relationship and find out if the student is struggling with a concept or has developed a misconception.
After our lesson in the high school, a big topic we discussed was that students were not engaged when the proof was being taught. Many of us believed that the reasoning for this was because proofs are boring and it is straight lecturing with the thought that perhaps because we were not their regular teacher that they did not feel the need to take notes. In addition, this was not a lesson that aligned with what they were currently learning so that probably was a factor as well. However, it got us talking about how to teach proofs and the realization that an average high school student does not have the same curiosity for understanding mathematics in the way that we as math majors do. As future math teachers, we know that proofs are an important part of mathematics and so we need to develop a way to teach proofs so that students care and are interested in proving math. I did some research and found an article that discussed 10 things that a current teacher wished she knew about teaching proofs before she actually was doing it. The link is: https://www.nctm.org/Publications/MathematicsTeacher/2009/Vol103/Issue4/Ten-Things-to-Consider-When-Teaching-Proof/
I will list all of the ten things to consider, but will only focus on a few.
- The research on the van Hiele levels is pertinent to proof
- Think about proof as a problem solving activity
- I could play a more active role as an advocate for reasoning and proof throughout the curriculum
- Be explicit about the purpose of a proof
- Be explicit about the structure of a proof
- Take time to read and study NCTM’s Standards publications while teaching geometry
- Logic should be explicitly connected to Euclidean proof
- Wait time is critical for creating space for student involvement
- Students should conjecture, not just prove
- It is important to teach proof not just theorems
First, I would like to focus on number 3. It is important that teachers are constantly teaching reasoning and proof in their classrooms even when it is not a “proof-based” class because this is teaching a way of thinking about mathematics as a bigger picture. They will develop skills of thinking deeper and, in turn, understand the mathematic much more and deeper than they would have otherwise. If students have never been exposed to this kind of thinking prior, they will become very frustrated when they begin to learn proofs because it will require a math mind not just a math memorization method.
Secondly, I would like to discuss number 5. Many students can develop the structure of a proof or pick it up, but there will always be students that will struggle with this aspect and though it may seem small in comparison, how can a student develop a proof if they do not understand the structure. I did some more research on this aspect and laying out proofs when you teach them and found this article: https://www.nctm.org/Publications/mathematicsteacher/2010/Vol103/Issue8/ProofBlocks_-A-Visual-Approach-to-Proof/
This article discusses using proof blocks as a better method of teaching proofs. A good quote that I found in this article explaining the reasoning behind proof blocks is, “When using ProofBlocks, students find it natural to work from both ends of their attempted proof, breaking it into smaller pieces and searching for the information they need to make logical connections. ProofBlocks is an inherently visual approach that lends itself to the use of manipulatives, thereby addressing the needs of visual and kinesthetic learners, who are often neglected when other forms of proof are taught” (Dirkensen 571).
While students still have to develop what goes in every space, they are guided or at least feel guided to make the connections themselves from step to step. They help students stay on the right track and see the connections of what theorem leads to what and how to use these previously learned theorem to prove new theorems. It is a good way for students to visually see the structure and layout of the proof.