MOEMS, AMCs, and MATHCOUNTS - those are the three math competitions that our lower school students participate in as part of their math class. For most new students to the school, that first math competition is quite more difficult than they anticipated. Even though we let them know not to expect 100% or that it’s different, it’s still a shock when they earn a 2 out of 5 on the MOEMS or an 8 out of 25 on the AMC 8.
We tell our students not to worry about how they do, that a 2 out of 5 is great for a first attempt in a MOEMS competition, and an 8 out of 25 for an AMC is also great! We’re looking for improvement over time, and the problems are difficult and like problems they are not used to solving. The problems between the three competitions are also different and challenge different skills. Speed matters with the AMC and MATHCOUNTS competitions, whereas speed is not a critical component of MOEMS.
Some of our students expend significant effort working to move as quickly through the math curriculum as possible. They view it as a ladder to climb to get to a certain level of math by a certain age. When speed through the math curriculum is achieved through significant effort, we prefer that a student meander a bit and explore the world of competition problems. For students for whom there is a gap in the level of math curriculum, and their competition results, we find that high-school math and science classes are difficult. We have seen that when achievement in competition problems is consistent with the achievement in the curriculum, students are better prepared to manage the complexity of their math classes in high school.
I recently watched a sweet video about two hypothetical children, one who has had practice learning how to learn, and one who hasn’t had that experience. This video spoke to me on so many levels as I saw the struggles that many of the children that I have worked with have experienced. I love math competition problems because it forces students like Susie to stop and struggle with problem solving. The benefits of learning the skill of how to learn is applicable to more than math. For our students, where math is traditionally a strength, this gives them the perfect opportunity to practice learning, practice being uncomfortable, and practice not intuitively knowing the answer.
How to Practice
So, can you practice for the competitions? Yes! For MOEMS, there are practice workbooks with problems like the MOEMS competitions. For parents it is something that kids can work on on their own. The answers are in the middle of the book so that the child can check his/her own work. If they get the problem wrong, then they can go back and rework the problem and check their work. If after reworking it, they still don’t understand, they can go to the back of the book and read the solution. After reading the solution, I recommend trying to resolve it again without looking at the solution. Art of Problem Solving also has Beast Academy, which is an online program that also has the option of books. The problems cover more curricular material, and logic. For AMC and MATHCOUNTS, there are past exams that students can take and practice.
I think of math competitions the way I think of sports. In this week of the Winter Olympics and the Super Bowl, I see athletes compete at levels that I can’t imagine, and it looks doable when watching it on TV - that is until I strap on the skis and fall all the way down the hill or imagine a 280 lb. man running towards me with the objective of knocking me down, for fun. The athletes we cheer on TV started somewhere, and it was fun. They also worked hard, training, competing, and pushing themselves. For students and math competitions, I would like students to treat the competition like we do sports. I once heard a fencing coach, Bill Grandy, give the best advice as kids were determining whether to compete for the first time, “You have to set goals. If you walk into that first meet and your only goal is to win, then you’re not ready to compete. Set smaller goals that are challenging, but doable.”
For a student who has never practiced or taken a math competition test before, walking in and expecting to get a perfect score is the equivalent of winning the fencing meet. I would love to see students participate in the first competition and use that as a baseline, then determine their goal for the following year. Goals could be, improve my score by 3 points or have enough time to try all of the questions. The goals should be based on how much time they plan on practicing. This practice will not only also translate into a different type of understanding of the curricular material, but will also help students with reading comprehension, speed, and creatively thinking about math problems.
Do you want to read more? Here are some additional resources that I have found helpful:
The teaching of math is fascinating. It is fascinating and complex. More than any of our other content areas, we have learned so much from our journey. A few years ago, at an MIT Club dinner, the gentleman next to me asked “Have you figured out how to teach problem solving?” My answer was quick, and I was excited to hear his response, “No, not yet, do you know how?” Alas he didn’t, and it led to a fun conversation about the teaching of math. Problem solving is at the core of math, and we’re still working on figuring out how to teach it to a broad range of students. Our experience has been illuminating, and a fun challenge to tackle.
When we designed the school, we read the research and work done by Patrick Suppes at Stanford University’s EPGY and by Julian Stanley at Johns Hopkins’ CTY to inform our approach that allowed students of varied math levels to progress at their own pace so that they could reach mathematically challenging material that would keep them engaged. Math, unlike other subjects taught in school (with the exception of foreign languages), builds on itself and requires a mastery of prior material to fully understand the new material being presented. In theory, self-paced math using adaptive technologies appears as a perfect solution. In practice, we found that it works for a smaller percentage of the population than we expected from the research, and even for those students for whom it works, they prefer the interaction with a person.
The beauty of our school being a 4th-12th school is that it’s like a longitudinal study; You see how choices made in elementary years affect performance in high school. Here are some of our findings:
For today, I will expand on this last bullet point, and we’ll visit other learnings in later posts. We have found some common areas of struggle in math which include computation errors, copying mistakes or not showing work, anxiety and freezing when working on a problem, and not understanding what the problem is asking. This is not an exhaustive list, and each could be the subject of its own post, but we’ll briefly explore each one.
An observation in teaching that took me by surprise, that shouldn’t have, is that I always need more time. I feel that I could fill an entire year of school with just one area of exploration, and we could have great fun exploring. Next year, we are adding an extra math period for math that will give us time to explore additional topics not covered in elementary and middle-school math curricula, such as set theory and proofs. That extra time is also being built in so that students have an opportunity to explore how they learn math. See you next week as we continue to explore math!
Teaching Tuesday: Don’t Fear the Red Ink: The Engineering Design Process in History Papers
About six years ago, I took my son to a summer seminar about The Lord of the Rings. What greeted me (ahem, us) were hundreds of books related to J.R.R. Tolkien. There were the books he wrote, books he read, books about the books he wrote, books about his invented languages, books about his life, books about Oxford, books about the Inklings, and many more books. It was at that moment that I wondered, “Why does my son get to have all of the fun, while I sit at the local Starbucks working while I wait for him?” Being a teacher, I knew that having an adult in a class can offset the balance of a class, but it never hurts to ask.
See, I love books. I love books, shoes, and purses, in that order. I couldn’t walk out when all of these books were there and were going to be explored -- without me. After quickly figuring out that I could do the work I was supposed to work on at night, I asked my son if it was ok if I asked to sit in the class (I didn’t want to embarrass him) and when he agreed, I approached the teacher. I told the teacher that I would pay for my registration, and I promised to not say a word. I just wanted an opportunity to sit in a corner in the back of the room and learn. I promised to be a really good adult. Let’s just say that the class did not disappoint and after learning what is philology, I am now a proud owner of a 20-volume set of the Oxford English Dictionary, which I have been able to collect as libraries have sold off their copies. If you know Tolkien, you know why that is relevant.
That teacher was Mr. Gardner, who is now one of our high school history teachers at Ideaventions Academy. After sitting in that Tolkien class, and the next one he taught on the Vikings, I realized that I had a gaping hole in my education. I enjoyed history when I was in middle school but did not take any history classes in college because I was afraid of taking the academic risk. The level of academic writing required in college history classes was too intimidating. I didn’t want this for our students, and we are so thankful that Mr. Gardner agreed to come and share his expertise, knowledge, and passion for history with our students.
Why research-based history papers?
We believe that learning how to write research papers is a multiyear process and no student shows up in high school or college knowing how to write one. It is our job as educators to expose students to long-term assignments where they are challenged to write a variety of papers, and history is a content area where they can develop the skill of writing research papers.
As a math and science school with a fair number of students who aspire to be engineers, we emphasize the importance of being able to communicate both orally and in writing. As students enter high school, we invite them to challenge themselves by taking one of our intensive history classes, where they will experience the conditions to develop grit and practice perseverance. Students in this class are exposed to college and post-college level texts, learn how to pick a topic, learn how to form a thesis, learn how to read complex texts within a timeframe, and learn how to write a paper that is organized, tells a story, and uses precise language and dates.
The class itself is split into two sessions - the history lesson and the writing workshop. In the history lesson, students are learning about the history that we are studying that year. The papers are done outside of class and use what they are learning in class as a way for students to figure out what area of what we’re learning about they will focus on in their paper. The writing workshop is where we work together on selecting a topic, finding books and journal articles, as well as reviewing outlines and drafts, if ready. In the writing workshop we introduce students to Turabian’s Student’s Guide to Writing College Papers, we hold technical lessons on the Chicago Manual of Style and learn how to footnote a paper. Through discussions, we also learn what goes in the body of the paper, or what is better suited for an appendix, and continue building our glossary of “vague words” that may not be used in papers turned in for this class.
How do we do it?
It’s a tricky dance between high standards and encouragement and support. The assignments include two short papers and a term paper each semester, and the topics include a critical book review, biographical essays, thesis-based research papers, and analytical essays. We like to give students a guided choice in determining what or who they will write about, as we have found that if a student is interested in the topic, they will be more motivated to work on their papers.
The paper is the final product, and part of what we teach students through working on these long-term assignments is how to plan and manage their time by providing long-term deadlines with the scaffolding of intermediate deadlines. We break up the assignment into two intermediate deadlines, topic selection and outline, with further suggested deadlines. We also recommend dates for when a first draft should be complete, but it is a suggested date, rather than a required date.
The Engineering Design Process in a History Class
What I have learned in teaching students how to work through this process is that it is a process broken down into subprocesses, and this process has many similarities to the engineering design process or the scientific method, which is a description our students understand.
Brainstorming: In picking a topic and researching sources, students start by brainstorming ideas for their paper. They then determine the feasibility by researching the available sources and evaluating whether the topic is too broad or too narrow.
Research: The reading and research phase is probably one of the areas where the most experimentation happens in class. Some students spend substantial amounts of time reading and taking copious notes, running out of time. Others barely take notes, then have to spend hours rereading. We learn that different types of notes work for different types of topics. We have experimented with highlighting, summarizing reading sessions, taking notes on post-it notes, taking notes on note cards. It’s a messy process and yet, we learn so much.
Design: Finally, students get to the point where they organize their notes into an outline. They learn that the title is more important than they realized and that everything in the outline needs to support the title and their thesis. They learn that the organization of an outline will help them tremendously when writing the paper - spend some time on the outline (design) and the implementation (writing) will be faster. Early on some students try to take shortcuts and try to write the outline while still reading based on what they think they will find. They also learn that there is a balance between an outline that is a collection of detailed notes (too detailed) and an outline that was thrown together 30 minutes before the due date (not enough detail). We also can’t forget to think about the introduction and the conclusion.
Implementation: Now, we finally get to writing the paper (implementation). Many people would be happy writing that first draft, and turning it in. We ask students to work through multiple drafts and classmates who have taken the class before share techniques that they have learned. To encourage working through the process rather than the grade, we allow students to submit a rewrite of their paper and earn an A- if they weren’t happy with their first grade.
Testing and Iterating: Students who take us up on the rewrite, apply the feedback that they receive, and learn from it, begin to see improvements in the next paper. The feedback they receive is extensive, and students learn that red ink is not a judgement on their writing or their worth as a person but take it as coaching and feedback meant to help them grow academically. When we explain that a teacher will only spend that much time giving feedback, it’s because he cares. In this dance of excellence and support, we also allow students to drop the lowest grade for one of the short papers.
We ask students that take the intensive history courses for at least two years to compare their papers from the end of year two to the first paper they wrote. They are shocked at how much they have grown. One student said, “That first paper was embarrassing.” They also experience how much faster they become at writing papers and learn about themselves as researchers and writers. The papers are also not as scary as they were at first. I wish all students in high school had the opportunity to spend this much time learning how to research, write, and communicate.
“There will come a time in most students' careers when they are assigned a research paper. Such an assignment often creates a great deal of unneeded anxiety in the student, which may result in procrastination and a feeling of confusion and inadequacy.” - Purdue Online Writing Lab
Our goal is for that time to be in high school and through the support afforded at a school like ours, to avoid the feeling of inadequacy, and build a feeling of empowerment and confidence. If you want to take a look, here are papers that students wrote about The Great War and World War 2 and compiled them into a website. We're still working on sharing the writing from other topics from 2000+ years of history that we have covered in our classes. In the meantime, enjoy reading what our students have written about this particular time in history.
Juliana Heitz is co-founder of Ideaventions Academy and is very excited to share the thinking behind the Academy.
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