Hello faithful readers and purveyors of this blog! I appreciate the comments and kind words concerning this site and its support as a resource for teachers and students. I'm currently deciding whether to continue on updating this blog or migrating elsewhere now that I am no longer in the Six Nations District Numeracy Teacher position.
Rest assured, I do not plan to wipe this blog off the interwebs, but may open up a new url/blog/address that more accurately fits my interests as an OCT professional. These include current math initiatives and progressive practice, the effective and fluid use of technology for education for the 21st Century (not old models on new devices), and cultural topics relating to First Nations, Social Justice and Equity. You may find the odd topic outside of these areas, but that's what I plan to continue sharing in the future.
If you have an ideas or comments that may help inform the future of my digital presence, please leave a comment below. In the meantime, this blog is about to go dark with few if any updates until a decision is made.
As I'm sure you've probably already seen or heard in the news that today marks the 50th anniversary of Dr. Martin Luther King Jr.'s "I Have a Dream" speech. Why not take a moment to use our district's BrainPOP resource account with your class to learn a bit about the historical event, by clicking here?
Not only is this blog post that I'm sharing with you, (found here), entirely brilliant in the way it takes a textbook problem and explodes it (or deconstructs it) to provide a richer learning experience for the students, it is also extremely funny and entertaining to read.
I have added the website to our list of awesome math websites on the sidebar. Enjoy!
The figure above shows a puzzle called OkiDoku. Prof. Dani Novak & Prof. David Rosenthal, Ithaca College
Many people enjoy working on grid puzzles as small, quick challenges of their mathematical and logical skills. Here is one you may not have seen, the OkiDoku. How does it work? Looking at the grid above, try to find four different numbers and put them in these 16 squares in a way that will satisfy the following two conditions:
Each of these four numbers must appear exactly once in each row and in each column.
The blocks with thick borders are called cages. Each cage shows a target number and a mathematical operation. The operation applied to the numbers in the cage should produce the target number. For example, there is a cage in the first row with a target number of 20 and a mathematical operation of multiplication. The puzzle solver should put three numbers in the cage so that the product of these numbers is 20.
Give it a try.
The most famous of all grid puzzles is the Sudoku, a logic puzzle found in a vast majority of newspapers. Some of you may have encountered a mathematical grid puzzle called KENKEN® that appears in more than 100 national U.S. publications. KENKEN® was invented by Japanese teacher Tetsuya Miyamoto and introduced in the United States by NexToy, Inc.
Professors Dani Novak and David Rosenthal of Ithaca College have created a similar puzzle called OkiDoku and used it to make learning math an enjoyable experience. Several other grid puzzles appear at a popular puzzle website called ConceptIsPuzzles. On the average, a whopping 20 million puzzles developed by this website are solved every day by adults and kids around the world. Clearly, there are many people who choose to solve grid puzzles as a recreational activity.
Motivation to Learn
Because many students enjoy working on these puzzles, they can be easily motivated to adopt learning strategies that will improve their puzzle-solving skills. Grid number puzzles provide strong intrinsic motivation to solve for unknown numbers from a handful of clues. As many math problems have a similar form, students who enjoy solving these puzzles can develop positive attitudes toward other forms of math in non-puzzle contexts as well. I have taught puzzle math to students in grades three to six. In these classes, I have found that students show a significant positive change in their attitudes toward math after a year of recreational math activities. Professor Harold Reiter of University of North Carolina, along with Professor Novak of Ithaca, also found that teaching puzzle math has a positive influence on student attitudes towards math.
Educational Value of Puzzles
Even without supervision, students can learn to be creative and persistent after working on many hard grid puzzles. In my classes, I augment self-exploration of grid puzzles with a guided exploration that teaches problem-solving, reflective learning and algebra techniques.
Let me illustrate this with the puzzle shown at the beginning of this post. Even though we can use the creative problem-solving and logical reasoning approaches that we usually use in solving puzzles, we can progress a bit faster on the above puzzle with some help from algebra.
Suppose the top number in the 11+ cage is x and the bottom number in the same cage is y.
We know that each row has to have the same four numbers. So the product of all numbers in each row is the same. As the product of first three numbers in the first row is given to be 20 and the fourth number is x, the product of all numbers in the first row is 20x. As the product of the first three numbers in the second row is given to be 35 and the fourth number is y, the product of all numbers in the second row is 35y. As the product of all numbers in the first row is the same as the product of all numbers in the second row, we know that 20x= 35y. We have also been given the clue in the 11+ cage that x + y = 11.
So what we have is a set of equations:
20x = 35y x + y = 11
This is now an algebra problem. The use of algebra to solve a problem that students are deeply engaged in allows them to appreciate the power of algebra and also provides a strong motivation to study algebra.
I have taught a yearlong extracurricular class that covers a variety of math problem-solving techniques and algebra in the context of grid puzzles. These techniques include:
Making a table
I have written three books that can be used to learn problem solving with grid puzzles. Additional techniques on grid puzzles are described at the Math Olympiad site.
Effectiveness of Puzzle Math in Improving Student Performance
In my classes, I teach math problem solving in the context of puzzles, letting students practice their skills with puzzles and finally encouraging them to ensure their mastery by applying their skills in other math word problems. I have found that students in my class show remarkable improvement in their math problem-solving and learning abilities. However, skill improvement among students varies significantly depending on other factors such as the amount of effort students put into doing homework, as well as student aptitude. In a national Noetic math contest, a majority of students in my puzzle math class showed significant improvement in math skills. Two students were perfect scorers, and a significant percentage of students won national recognition.
While puzzle math is valuable for achieving certain educational objectives, it may not be the best choice in all educational situations. For example, if students and teachers have short-term objectives to achieve the best results on particular tests with specific types of questions, then practicing questions similar to those on the tests can help them achieve those objectives -- not the approach described above. Similarly, teachers may want to emphasize the relationships of math concepts to real-world applications. Again, puzzle math would not support this objective.
In short, teachers will find the puzzle math approach useful to help achieve certain objectives but not others. However, there is no question that teaching math problem solving in the context of grid puzzles, long regarded as a recreational activity, is effective in cultivating students' interest in math and in improving their problem-solving, reflective learning and algebra skills.
3 Ways To Not Screw Up The Curriculum Mapping Process (view the original source here)
The school year isn’t a series of sprints, but the way you forge your curriculum can make it feel that way.
The most common way of structuring how you teach is by first assembling standards into units, then those units into lessons.
You may use a backwards-design process (popularized by UbD and Grant Wiggins), where you start with what you want the students to understand, then decide what can act as evidence of that understanding, then finally design an assessment that provides the best opportunity to uncover what students know.
This is an entirely rational response to the huge workload of surveying, harvesting, bundling, and distributing the breadth of academic standards in your content area.
This is especially true if that content area is English-Language Arts, which has no less than six separate sets of standards in Common Core menu. That means dozens and dozens of standards ranging in complexity from “spell correctly” to “Analyze multiple interpretations of a story, drama, or poem (e.g., recorded or live production of a play or recorded novel or poetry), evaluating how each version interprets the source text. (Include at least one play by Shakespeare and one play by an American dramatist.)”
Organizing makes sense. Sequencing is a type of organization—thus Scopes and Sequences and Curriculum Maps based on given academic standards. These documents function in a lot of ways, primarily in ensuring all of the content gets covered, and creating the possibility of a common experience for teachers so that data and instructional resources can be shared.
The challenge comes when that organization creates artificial barriers and awkward pathways through content. In pursuit of “getting through it all,” it’s easy to encourage bad thinking habits, and worse, provide misleading data about what students actually understand.
Imagine for a moment each of your units. Whether you use genre-based units (a “poetry” unit, an “linear equations” unit, etc.), thematic units (where learning experiences are planned around themes and thematic questions), project-based learning, or some mix of these approaches and others, not all content is equally important.
So how can you keep from screwing up your curriculum map? Or revise the one you were thoughtlessly handed 3 days before school started, taking lemons and making lemonade?
Recently, the concept of “power standards” has surfaced, recognition that not all academic standards are created equal. Some standards are naturally more interesting than others, more enduring than others, more complex, or can be leveraged to aggregate other standards and related content together. For example, in English-Language Arts, a standard involving explicit and implicit themes might be considered a power standard due to its ability to involve other standards or content, including: author purpose, audience awareness, tone, text structure, theme vs thesis, and others.
2. Use Spiraling
It makes sense then that if certain content is “more important”—for any number of reasons—that content should be “spiraled.” We’ll get more into spiraling in a separate post, but essentially spiraling in curriculum is the process of embedding critical content throughout the year. This usually means that at the beginning of the year, this content is delivered at lower levels of Bloom’s taxonomy. So if we take the aforementioned example of “theme,” in August theme will be defined, examples will be given, and early analysis will occur. By mid-year, students will be analyzing themes of simple texts more closely, and begin analyzing the themes of more complex texts (and digital media), and by March, students will be analyzing complex themes of complex texts.
3. Diversify Assessment Forms
The process of responding to the personalized learning needs of your students likely begins with assessment. Even if you don’t differentiate the content, processes, or products of learning, simply altering how you assess understanding can go a long way towards truly personalized learning for students.
Whether you simply offer students choice in assessment forms—multiple choice versus concept-mapping, short response versus student conferencing, an exit slip versus a journal entry—the more variety and choice you can build into assessment, the better you can protect students from the problematic curriculum mapping practices than can sabotage student academic performance.
Testing is not engaging, but assessment can be.
In seeking to cover all of the content, do all that you can to avoid the phenomenon of sprinting through unit after unit. The content that will serve both you and students most powerfully likely needs to be prioritize and spiraled throughout the academic year.
If you can also diversify assessment in your classroom—offering student voice, choice, and the ability for students to prove what they understand—and the depth at which they understand—in a variety of ways, the better the chance you are able to meaningfully respond to the personalized learning needs of your students.
Yet another list of 10 for you to countdown the days of summer. Take one a day for 10 business days while you consider using 20% time (a.k.a. #geniushour for you Tweeps out there) in your classroom in the fall... Originally posted byRoss Crocketton
“Here's a radical new idea you may not have heard of—20% time. With this concept, you devote that percentage of class time to allow students to discover and learn the things they want to, and in the ways they want to learn them. Sound crazy? I urge you to try it on in your own classroom. Below, AJ Juliani provides us with a list of reasons why it's worth considering as a classroom teaching and learning strategy.”
If you haven’t heard of 20% time in the classroom, the premise is simple: Give your students 20% of their class time to learn what they want. Yes, that’s it. Below is a list of the 10 reasons you should consider 20% time in your school, and you will not regret making that choice!
1. You will join a great community of learners
When I first did the 20% project with my students I didn’t have a community of teachers or learners. Within months that changed as a number of great teachers before and after me started to share their 20% time stories online. The largest active group is the Genius Hour teachers (inspired by Daniel Pink) who have #geniushour chats and a great Genius Hour wiki. Get involved and see what others have done!
2. You will allow students to go into depth with a topic that inspires them
One of the major issues we face in schools today is covering a wide breadth of information, instead of allowing students to get a real depth of knowledge. Students using 20% time are able to delve into subject matter that means something to them, often times taking their free time at home to learn more. Isn’t this something we should be promoting at all levels?
3. There is so much positive peer pressure
When students in my school have their pitch day, they get to share with the entire class what they are working on. Publicly announcing what they are trying to accomplish makes the goal real. Students get to see what their peers are working on and want to make sure their project stands up to the rest of the class. Regardless of a grade being attached to the project, this makes for students going the extra mile.
4. It relieves students of the “game of school”
Too often our students complete assignments for the grade. They go through the motions to receive an external pat on the back (or pat on their transcript). 20% time takes away the “game of school”. It brings back the love of learning for learning’s sake.
5. It’s fun!
Randy Pausch famously said, “If you think you can’t learn and have fun at the same time. Then I don’t think you have a good understanding of either.” Without a doubt it is the best time of the week. Listen to how one Genius Hour teacher share her students’ excitement for Genius Hour to arrive.
6. Your class will be covering all types of common core standards
It doesn’t matter if you teach elementary, middle, or high school. The genius hour and 20% time projects cover multiple common cores standards. We’ve had teachers propose this type of learning to their administration back by awesome research. Remember, the community will help if you are fighting a battle to get 20% time started at your school.
7. It’s differentiation at its best
Students are working at their level, and as teachers we should be helping to challenge each one of our learners at their best pace and ability. Because each project differs, students are not bogged down by following the same steps as their classmates. The entire class is learning, but it is truly differentiated.
8. You learn by what you do, not by what you hear
Experiential and challenge based learning puts the mastery back into the student’s hands. We provide guidance and pushes along the way, but they are the ones “doing” and “making”. Confucius put it perfectly: “I hear and I forget. I see and I remember. I do and I understand.” Let your students make and they will understand and thank you for the opportunity.
9. It is a perfect way to model life-long learning
I did the 20% time with my students and took it upon myself to learn how to code and make an app from scratch. I failed to make that app. But my experience learning how to program left me with a whole new perspective, and was a teachable moment about what we call “failure”. There is no real way to fail a project in which “learning” is the end-goal.
10. Your students will never forget what it felt like to create
Have you seen Caine’s arcade? It started out as a little idea and now Caine has inspired hundreds of other kids his age to create something unique. When you create a product, it becomes part of who you are, and there is a “care” involved that we just never see with multiple-choice tests. What would you want for your child?
This is the most important time to be in education. It is the most important time to care about education. It is the most important time to impact education.
Now, more than any other time in the past 100 years, education seems on the verge of a paradigm shift. You see, for the past century, most of the educational change has been “doing old things in new ways”. Today, we are beginning to see educators, educational institutions, and educational companies do “new things in new ways”.
My challenge to you as a teacher is to allow your students the freedom to learn what they want. That’s what 20% time is all about, and that is why it is so successful.
Continuing our theme of lists of 10s, take 10 business days to consider these 10 practical ideas for better project-based learning... Originally posted byRyan Schaafon
“Project-based learning is a powerful approach to teaching. This approach allows students to learn content in a manner similar to how they will work during the rest of their lives. PBL provides teachers with easy avenues to engage students with real world problems that are relevant to the learning community. It is possible to teach multiple subjects simultaneously and make connections outside the classroom. This extensive list provided by Jennifer Nichols in this article she wrote for TeachThought makes a convincing argument for the merits of project-based learning in any curriculum. ”
Teachers are incorporating more and more projects into their curriculum, allowing for much greater levels of collaboration and responsibility for students at all levels. Project- based learning is a popular trend, and even teachers who don’t necessarily follow that approach still see the benefit to using projects to advance their students’ learning.
Projects can be wonderful teaching tools. They can allow for a more student-centred environment, where teachers can guide students in their learning instead of using lectures to provide them with information.
The increase in classroom technology also makes projects more accessible to students. Research no longer requires a trip to the library, and displaying information no longer requires a poster board. Instead, students can access endless amounts of information with a few clicks, and create all kinds of creative final products (such as slides, videos, cartoons, ebooks, blogs, websites, graphics, and much more).
Despite general agreement about the benefits of using projects and project-based learning in general, it must be noted that all projects are not created equal! It is quite possible to create projects that remove creative ability and control from the students and places all the power of decision with the teacher.
This may happen fairly often because teachers are wary about being able to assign grades to the final assignments handed in to them by students. If all students aren’t given the same components to work on, with similar topics, and the same final layout to create, then how can the projects be accurately compared by the teacher? In short, they can’t – but also, they shouldn’t!
Students do not need to be compared against each other, but to the standards they need to achieve for their level. How each skill is demonstrated can differ from one student to another, yet each student can succeed nevertheless. A teacher who knows the program will know what skills each student needs to acquire, and present them with situations to help develop those skills. A teacher can also gauge whether a student has developed each skill regardless of the way they choose to create and present their project.
When students are engaged and interested in the work they are completing, the final product will be much better than when they feel forced to complete a task.
…but how can you make sure that the project you assign is engaging to as many students as possible – if not to all of them?
Here are some great tips to keep in mind when putting together your next project.
1)Have students work in small groups or pairs whenever possible.
Don’t underestimate the power of collaboration. Working alone can be great at times to place a student’s level of ability on their own, but it can be frustrating to a student when they run into parts that they are less adept at. Peer support can help keep things running smoothly, and also help students to build the skills that they are lacking by learning from each other.
If groups are too large, students are given the opportunity to shrink back and leave the work to others, but pairs or groups of three allows everyone to share input and really take on a role within the project. Don’t be afraid of assessment from projects, you will be able to tell how each of your students are developing by maintaining a constant presence in your classroom and observing/interacting with your groups.
Being able to work together will definitely keep students more engaged in the work, especially since they become responsible to each other and to themselves for the completion of the project. There is less chance of students giving up or giving in mediocre work when they are being counted on by peers and are having fun.
2) Choose skills to be worked on instead of specific topics.
The goals of education focus on helping students to build the skills that they will need for their future. These skills involve being able to collaborate, write well, read between the lines, infer meaning, organize information, find solutions to problems, research effectively, and learn about their place in the world.
When forced into specific topics, students are limited in their ability to be creative and to focus on learning information that they find relevant to their lives. Instead of asking students to all complete projects on an animal, for example, why not decide on a few target skills and build a project guideline that can be used for many different topics instead? That way, students can focus in on something they would like to learn more about, while following your guidelines to make sure that the skills you are targeting are being developed.
If you want students to define a set of problems associated with something and work together to try to find plausible solutions to those problems, there is no need for every group to be working on exactly the same topic. This will also make things much more interesting when it comes time to present the projects, instead of listening to each student’s version of the same thing!
3) Give students guidelines that allows for individuality.
After choosing the skills or content that you would like to be the focus of your project, build guidelines that support student individuality and creativity. Instead of making a list of specific questions with specific answers (such as ‘what is the habitat of the grizzly bear’ questions), lead students towards more open-ended answers in your guideline.
Using questions such as ‘list three facts that you found surprising while researching the topic and explain why they surprised you’, ‘based on the information that you gathered, explain why you think ____ happens’, and ‘explain what the top ten things people need to know about your topic are in order to understand it well’ can really lend themselves to multiple subjects.
Your guideline should list the skills that students are working on, so they are aware of them and can actively work on developing them. If you want students to learn about democracy and how the government works, as well as to develop their problem solving skills, then telling them to build their own country – similar to the USA or Canada in structure and government – but with their own flair added in, can be an engaging way to do it. Allow for some crazy bits included in their constitution, or even elections where voters submit to X-rays instead of bringing ID.
In order to complete the project, they will need to research the government you want, and take it even further by using the information as a basis for their own creations.
4) Encourage students to take on different roles while collaborating.
In order to get all students involved in a project, don’t allow them to simply break it up and then put it back together after each student has individually covered a section of it. Collaboration in the real world involves being able to work together on each part of a task, while learning to compromise and solve problems as they arise.
We rob our students of some great practice when we split tasks! Depending on the needs of the project, you can have graphic designers, managers, organizers, researchers, etc. While one student would be named ‘in charge’ of graphics, for example, they would still be working with the input of the rest of the team – much like how adults collaborate on projects in the ‘real world’.
Encourage them to switch roles as needed, based on the strengths of their team, or on the skills each student needs to develop. No one student should always be ‘in charge’ or ‘approving’ all the work.
5) Allow students creative choice when it comes to the final result.
Do you really need that project to be presented on a piece of cardboard? If so, then make sure you have a good reason for it! There are so many ways for students to demonstrate learning, especially with the integration of technology, that it seems rather ridiculous to rob them of the chance to decide for themselves how to showcase their work.
What you really want is found in the content of the project, not in a piece of paper or cardboard. When students take ownership of the method of presentation, you are sure to be blown away with some extremely creative and innovative presentations. Allow them to make eBooks, videos, movies, animations, mind maps, skits, game shows, talk shows, newscasts, magazines, podcasts, blogs, or anything else they can come up with!
6) Change the way that projects are presented/displayed.
Even if every group in your class presents in a different way, you will be able to assess each and every one of the projects, based on the skills/content that students need to show you. Students will also look forward to presenting their project, and seeing the presentations made by others!
Also, instead of just pinning projects up on a board, or sending them right home after being presented, consider displaying them in more creative ways. Ebooks, articles, videos, and other media can be incorporated into a class website or blog, where other members of the community can access and appreciate the work.
If students know that their work will be shared online or in the school/community, they will likely be more excited about putting their best foot forward. When you know your work goes directly into the recycling bin once it’s finished… well, less effort tends to go into it!
7) Grade projects based on the targeted concepts and skills.
Create grading rubrics or charts for yourself that help you to focus in on the specific skills that you are looking for in each project. Since each group may have a different final format, you won’t be able to compare them with each other very well (which we shouldn’t really do in general when assigning grades). Students are supposed to be graded on their level of development when compared to curricular goals, not based on comparisons with each other.
Try to avoid assigning grades based on how great one group’s video was when compared with another group’s poster board. While one might stand out more, the other may just have better content!
Letting students know exactly what you will be looking for beforehand will make it much easier for you to see what you need, as they will usually make sure to show it to you!
Besides, let’s face it, if a university professor didn’t explain exactly what was required of an assignment, but graded you based on what they were mysteriously looking for, you would be frustrated and have a hard time doing as well as you could in that class. Our younger students appreciate the guidance too!
8) Consider cross-curricular activities and/or work with another class.
Projects tend to be more engaging if students have the chance to immerse themselves in them as much as possible. Seeing similar content appear in multiple subject areas helps to reinforce what students are learning, as well as make the learning more relevant to them.
In many cases, mathematical or scientific learning can be added to English projects. History is another subject that lends itself well to cross-curricular projects.
Working with another class can also be fun for students. With today’s technology, it is even possible for students to collaborate on projects with classes in other schools – or even in other countries. Doing this definitely helps to prepare students for their futures, as we often find ourselves working/collaborating with coworkers in other departments or cities. It can add an extra challenge to organizing work and getting things done efficiently!
9) Give the project a purpose beyond the classroom.
If possible, try to build connections to the outside world into projects. If students can work on something that will directly benefit the school or community (such as planning and implementing a fundraiser, or creating books/movies for a community centre or home, or even planning a special lesson for younger students) it can really help to build engagement in the class.
Knowing that your work will do more than get you grades – that it will actually be used to help people – can be a powerful motivator. Students will also help to motivate each other when they know that their work is important and useful.
10) Incorporate the project into the students’ digital portfolios.
While not all projects can directly impact the community, they can at least be used as evidence of learning in students’ working digital portfolios. The great thing about digital portfolios is that they follow a student as they advance through the grades and paint a picture of progress over time.
Once again, doing this can help curb the de-motivation of knowing that work will just be thrown away once completed. Incorporating student reflections and teacher/peer/parental/community feedback can also be a nice way to follow up on the learning that has taken place, as well as provide some future goals to work on in order to improve on skill development.
Beyond just tracking learning, the allure of being able to go and watch a video project you created years ago seems too good to pass up; I can remember a few of my own projects I wish I could see again!
Sir Ken Robinson was asked to curate a TED Talk list on education. Click here to see the Top 10 TED Talks he personally selected. Watch one every day for the next 10 business days and we'll be back with a project based learning post. Hopefully these posts and videos help you to stay inspired and fired up for a new school year in August! See you then...