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.
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/
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.
I found an article from the National Council of Teachers of Mathematics entitled Investigating Problem Solving Perseverance Using Lesson Study, and I found it very interesting. It tide together lesson, which is the main topic of our course right now, and it gave some really good insight on how to pose difficult problems such that students are going to struggle. The study then comes in on how to help get students get through that struggle, and to what degree of involvement should teachers have.
Through lesson study, a group of teachers conducted three different lesson studies at three different grade levels in order to study how well the students persevered.
The first group was a group of 10th grade algebra one students. The findings were that students would often get off task as soon as the teacher would not be next to their group. Some solutions they came up with were to assign roles to students, holding students accountable for progress for their groups work, and allowing students to only ask the teacher one question when the teacher would check up on the groups.
The second group was a group of 11th grade honors algebra two students. They seemed to stay more on task regardless of where the teacher was, but that could have been because the lesson was set up so that they received a new problem every ten minutes and had only ten minutes to work on it. A concern that this lesson brought up was that it was too scaffold-ed and did not allow for any point of open struggle or pondering. It was more of a follow a step by step thing to so, but the concept was a little more difficult.
The third group was a group of 7th graders. To the high school teachers’ surprise the middle schoolers were willing to work in groups and share ideas with other students. There was not as big of an issue in maintaining student interest at this level as there is in high school.
Perseverance was really looked at as students being willing to work through difficult problems, but needing reassurance at critical points in their work. By sharing out different strategies to attack a problem also shows a method of perseverance.
I really liked how the teachers in the article used lesson study as a point to look for perseverance. It shows the range of topics that lesson study can cover. I would like to use lesson study for checking for skills and point like perseverance and other topics. Lesson study is a really good tool for the classroom. Also, teaching students how to persevere through these problems is important because it teaches kids that they can do mathematics and teaches skills that pertain to larger parts of life.
A growing interest in teaching with productive struggle is a hopeful sign in the math classroom. Productive struggle can lead to deeper understandings, connections, and motivation in students. So, why are some teachers still hesitant to use it?
Recently, I was able to present a lesson over the Law of Sines and Cosines in a local high school. Normally, these students learn in the traditional way of lecture. However, we challenged ourselves to create a lesson that caused students to struggle a little before they came to the solution. When we presented it, many of the students seemed disinterested and unengaged. I was worried about this when creating the lesson, but I did not think it would occur to the extent that it did. So, I have looked into articles that provide ways for a teacher to slowly introduce and transition students into lessons that incorporate productive struggle.
The article, Beyond Growth Mindset: Creating Classroom Opportunities for Meaningful Struggle, gives tips on what to do and what to avoid when teaching with struggle. Years of research has proven that student learning is enhanced when they have to be persistent to reach success–this concept goes all the way back to the educational reformer, John Dewey who “described learning as beginning with a dilemma.” One major key to a successful productive struggle lesson is to have the goal of the lesson be focus on getting students to have a deeper understanding of the material instead of just focusing on creating struggle. One common way to do this is through real world applications. For example, in the lesson we created for the geometry class, there was an activity that involved drones and using them to deliver pizza. Although this is not necessarily happening today, it is something that could take place in the near future. When this activity was introduced towards the end, there seemed to be a switch that flipped. Almost all of the students were working together to figure out the problem. And, they were able to recall previous math knowledge to help them solve the Law of Sines problem. This is another helpful tip when causing productive struggle–having students use math that they are familiar with can help produce productive struggle instead of frustration. Having the knowledge needed for part of the problem gives students a boost of confidence. However, students also need the time to realize these connections. As a teacher, you should be sure to give your students enough time to think about the activity before giving them any hints or guidance. This time is crucial for allowing students to make connections and deepen their understanding of the math at hand.
The article provides the following list of key elements for providing productive struggle:
- Determine timing and placement for productive struggle within the unit or curriculum—lessons that are “preparing students to hear something really important.”
- Align struggle activities with clear, specific learning goals.
- Design struggle tasks based on assessment of students’ prior knowledge and skills.
- Foster a safe environment that encourages student inquiry and exploration of important ideas.
- Use probing questions to solicit student thinking and provide strategic assistance to nudge students through their zone of proximal development–the zone of students’ thinking just beyond the level they can do completely on their own.
- Follow-up each struggle episode with carefully structured lessons that build on students’ ideas, address misconceptions, and help students forge new understandings.
- Assist students to reflect and articulate what they learned as a result of productive persistence.
In the future, these tips will definitely help to transition into a classroom with productive struggle much better than the geometry class I just experienced. For instance, I think that the students would have gotten much more out of our discovery activity for the Law of Sines and Cosines if we gave them more thinking time and created an environment that allowed students to talk about their ideas without fear of being wrong.
Last week I discussed nine ways to motivate students, as found in an article linked in that post. Early this week, Dr. Reins sent out an article about a 10th way to motivate students. The tenth way being promoting productive struggle (link: https://www.edutopia.org/article/promoting-productive-struggle-math) through bellringers. In the article, the author (a teacher) discusses different problems that he presents his students that have minimal words so the students have to figure out what the question is asking, and how to answer it. Within the article was a link to a TEDx Talk called “Teaching Without Words,” presented by Matthew Peterson. He talked about taking away words from math problems, and instead creating a computer platform full of games to teach math. A platform that contains no words. At first, I was skeptical of how this may work, mostly because all of the students’ learning then becomes on a screen and there is no tactile component. However, Peterson cited a study that they conducted about the impact of removing words from the mathematics classroom and replacing them with these games. It was conducted in five cities, and across the board the math proficiency rates almost tripled.
After reading the article that was sent out and watching the TEDx Talk, I began to think about how I can promote productive struggle in my own classroom, besides removing words from instruction. Although I believe that removing words from problems is beneficial for students, I think that in my future classroom I would like to implement removing words without turning to solely technology. Incorporating technology into the classroom is beneficial, but I believe that there are benefits to using tactile learning, so I would modify the original idea in that. Using ideas similar to what Giardi implemented in his school would be interesting for students, and a good way to begin a lesson with the students. Considering removing words from my instruction (of which there are many in mathematics textbooks), I began to search on NCTM’s website, I found an article (link: https://drive.google.com/file/d/1MolRN-jEtJPBiYVMJWWxX3g0tixHSObS/view?usp=sharing) that focuses on supporting productive struggle through communication with students. Although I have seen it before, the content within is still applicable and helpful. Specifically, the article discusses different communication “moves” a teacher can make to help provide productive struggle in the classroom. It provides three communication moves: assessing questions, advancing questions, and judicious telling. Assessing and advancing questions are clear in their implications, but judicious telling is subtly encouraging students to get onto the right path. I could implement these communication strategies in my future classroom to align with NCTM’s publications on promoting productive struggle in the classroom.
In terms of inciting productive struggle in a classroom, as we saw this past week in the lesson study, it can be much more difficult than we hope. The students in the class were unused to a teacher asking them to struggle in finding the answer to questions asked. Promoting productive struggle in the classroom is challenging for teachers, but the end result is students who have a deep understanding of mathematics, have developed applicable skills, and more.