Being more spatially-challenged in general, I always had trouble as a child comprehending concepts like mirror images, rotations, and geometry nets.
Fortunately, as a grew older, I learned that these are all just facets of broader concepts of scale and perspective. I’ve also benefited by recognizing their applications beyond mathematics–from art to city planning to interpersonal relationships.
So this week consists of a provocation to help our young learners begin with the big picture of scale and perspective, hopefully encouraging them to draw their own connections and conclusions.
The first is a fascinating video that lays out the entire history of the earth on a football field.
The second is a photo series by artist Matthew Albanese. He creates stunningly realistic landscapes using forced perspective, using materials from nutmeg to steel wool to fake fog. Head over to his site to view the collection of images, along with the incredible behind-the-scenes images and information on his process.
How do people use scale and perspective to help us see “the big picture?”
How does technology allow us new possibilities to show scale and perspective?
How do scale and perspective change the way we see the world?
What is our responsibility to use perspective in our lives?
How are scale and perspective connected?
How does perspective help us understand other people?
My question is this: even if teaching the trick gets students to pass the test and ace the class and get into the college–have we, as educators, truly done our jobs?
If we’ve never heard their creative approaches to making sense of math because we’re too busy telling them the right way to “borrow,” have we joined them in their learning journey, or are we scripting it?
If we just keep focusing our energy in helping them memorize, are our students ever going to see themselves as competent mathematicians?
Many of us have a love/hate relationship with math. And depending on your students’ ages, they may have thoroughly convinced themselves that they hate it or are no good at it. If that’s the case, we have all the more responsibility to help our students see the bigger picture and the true beauty behind the numbers, starting with our own attitude.
This is easily the most phenomenal mathematics video I’ve ever watched. Share it with your students to provoke inquiry and appreciation for math–and at only 1:41 minutes, don’t be afraid to play it again and again as the conversation deepens and understanding sinks in.
The Daily 5 and 3 for literacy and math: perfect for addressing some questions I’d had on inviting more student choice and ownership. Unfortunately for me, my school adopted it the very year I began my extended parental leave. However, I was thrilled when I was invited to mentor a student teacher that fall, allowing me to still test out the Daily 5/Daily 3 waters for myself. And after a few weeks, the students and I agreed that it was a worthwhile change.
Meanwhile, not everyone at the school welcomed the transition with such enthusiasm. Some worried about not spending enough time on spelling. Others worried about students squandering time. Others were simply entrenched in their existing routines. If you are considering either program, here are some tips to keep in mind to foster a smooth transition.
Allow a LOT of training time
This is no joke. Most students have learned “school” pretty well, but that tends to be more of a teacher-directed perspective. The autonomy of evaluating how they need to spend their learning time is going to be quite novel for most of them. Take each Daily 5 or Daily 3 choice one at a time, emphasizing not only stamina, but metacognition to support their ability to reflect upon their own strengths and needs.
Use status of the class–especially starting out!
One of the recommendations in the current Daily 5 book for monitoring which Daily 5/3 choices students make is roll call or status of the class. It enabled me to track their choices and to offer brief feedback so they could learn to really plan their time well.
Many teachers I spoke with felt it would be too time-consuming to call out each student’s name for their response. However, after a period of training on this process as well (we even timed ourselves to make it a competition), we were able to finish in under 2 minutes. Especially for older students, over time, you may be able to eliminate this step and let students simply move their name or picture on a choice board (such as the example below).
However you decide to track their choices, avoid the temptation to regularly assign them to stations. This eliminates one of the fundamental purposes of Daily 5/3, which is to foster students’ ability to determine how they need to spend their learning time.
Make the schedule work for you
Don’t be intimidated by the way blocks of time are outlined in the book. Interruptions to the school day are almost always a package deal, but the good news is that Daily 5/3 are designed to be flexible.
If the time you have available for student choices time is a bit shorter than ideal, add one more Daily 5 block (without any whole group time) during the day for them to choose another station to revisit and catch up on. See the example schedules at the bottom.
Don’t skimp on wrap-ups
Despite the flexible nature of Daily 5/3, don’t skip the wrap-up! This moment of reflection is invaluable both for you and students to gauge the progress, problems, and successes.
Stagger the mini-lesson one day and assignment the next
If you don’t have enough student choice times for all students to get to a station that includes an assignment based on the mini lesson, simply give the assignment the day after the corresponding mini lesson.
Make an assignments board
Simplify where students should look for Daily 5/3 assignments (and possibly a reminder on essential agreements) by designating a bulletin board or a corner of your whiteboard. See below for a great example.
Don’t drown their choices with teacher-centered worksheets
It may be especially tempting in Math Daily 3 to make each of the stations different kinds of worksheets from the lesson manual. However, keep in mind that one goal for Math Daily 3 is to foster more hands-on learning experiences. Both “Math by Myself” and “Math with Someone” are intended for games and exploring math manipulatives (see next tip). “Math Writing” is appropriate for students to show their understanding on paper.
Create a running bank of games/activities for math
As students learn each new game or math manipulative activity, write down the title on a sentence strip. Then, for Math by Myself & with Someone, you can just pull out familiar games for new concepts (or for review, especially at the beginning of a unit). Examples:
This teacher has prepared gallon ziplock baggies of games ahead of time for partners to play together. Her examples are geared toward grades 2-4, but the concept is great because it reinforces having a bank of games the kids are familiar with. This would be a great parent volunteer activity!
If the noise level is reaching a distraction for students in independent stations, seek out solutions as a class. For instance, they might find limiting the number of partners that can work during a block to be helpful.
EXAMPLES OF SCHEDULES/CHOICES FOR 2 DIFFERENT CLASSES:
I used to love Mountain Math and Mountain Language. The spiral review. The simplicity of swapping laminated cards each week. The security of knowing my students were practicing concepts that could show up at the end of year tests.
During Independent Study time, students would grab a fresh answer sheet and try their hand at weekly examples of 20 grammar concepts (ie, parts of speech, dictionary guide words, spelling corrections, syllables), and about 22 math concepts.
However, the summer after my second year of teaching, I began to doubt. Was it worth the sizable chunk of time spent every week? Did it help struggling students to improve? Did it help not-struggling students to grow? Were there better ways to help them with retention? Most importantly, what was the big-picture program design more about: students becoming better readers, writers, and mathematicians, or standardized test drill?
As a fifth grade team, we reflected, and came to realize that while it did have some merits, the program was an opportunity cost for better things. We scrapped it cold turkey and worked together toward more purpose, more thoughtfulness, more curative effort, and more reflection.
What Changed in Language Arts
I was already committed by that point to wrap-ups for most lessons, but I became even more acutely aware of their necessity. Wrap-ups became a golden time for connection-making and conclusion-recording. I began to be more mindful in helping my students highlight specific concepts that occurred naturally in our lessons.
With the extra space, I got a second large bulletin board installed on my wall, and designated one for reading workshop and one for writing workshop. As we shared our connections and defined new concepts (especially during wrap-ups), we would record and display them on our bulletin board throughout each unit. Not only did this serve as a helpful visual reminder as we built upon unit concepts, but the connections to grammar ideas became more organic–which resulted in greater student ownership and retention.
Independent Study Shift:
Our school’s practice of dedicating about an hour of independent language arts study time underwent a gradual transformation over the next few years as we worked to identify better ways for students to practice language arts while teachers met with small reading groups. Eventually, we realized that students could learn how to prioritize that time themselves, if only we gave them the tools to do so. And so we adopted the Daily Five, which helped us lay out a better structure in teaching students to make purposeful choices for how they spend their time. Choices included read to self, read with someone, word work, work on writing, and listen to reading. I loved the shift in the mentality even more than the shift in the program selection.
Mini, teacher-designed Grammar Practice:
We started to design and select our own mini-grammar practices wherever we noticed students could use extra practice. When I went on extended parental leave, this was still an imperfect process, but I was excited about the direction and potential for growth.
What Changed in Math
Because we did not rely as heavily on the Mountain Math program, things did not shift quite as dramatically in that subject. Our most tangible change was implementing mini formative assessment quizzes. This involved creating small, two to four question quizzes each day based on the previous day’s study, often throwing in one bonus review question. As a result, we became more deeply and continually aware of the class’ understanding, and became better equipped to course-correct as needed.
What Changed in Me
In the end, this was a story about shifting ownership–both for my students and for me. I became more aware my students’ needs because I did not just rely on a program to “cover” concepts. I became more confident in my students’ abilities to choose what mattered most for their own learning–especially as I searched out meaningful tools to help them learn how. The bar was definitely raised for us all, but I have found it to be one of the most worthwhile changes in my teaching career so far.
Have access to computers, but short on the 31 protractors you’ll need for today’s lesson? Or looking for more ways to inspire hands-on math exploration? Look no further than our list of top digital math manipulatives, teacher tested to be student-friendly and relevant!
This is literally a one-stop bank of easy-to-use manipulatives. Geoboards, number cubes, pattern blocks–the works! It also has all sorts of bells and whistles to construct inquiry-based environments including various workmat or game backgrounds, tools like a stopwatch, compass, and ruler, and even a print screen button so students can turn in their math thinking! This is definitely a must-bookmark link for math teachers!
Solve Me Mobiles takes the abstract nature of algebra and turns it into something students can concretely tinker with. Through trial-and-error, students balance the mobile by entering the values of hanging shapes. This is a perfect precursor to algebraic equations.
This is another library of manipulatives, with the additional feature wherein most of the manipulatives are gamified–there are tasks and “check the answer” components in most. Manipulatives include pattern blocks, transformations, circle graphs, measuring angles, prime factorization, and more!
This resource likely includes every type of math manipulative you could fathom, and then at least a few more. Just make sure you have Java downloaded and that students use it on a browser other than Chrome (since Chrome and Java do not mix at the moment). Alternatively, they now offer a computer app for purchase to bypass the Java issues.
Speaking of apps, ABCya.com has put out a free app with some of their top virtual manipulatives for fractions, decimals, and percents. Students will love camera feature, allowing them to project math concepts onto everyday objects.
After boring both my students and myself with largely direct instruction math for a couple of years, I decided to try guided math. The results? Increases in interest, one-on-one time, student initiative, and just plain joy in math learning.
Why Guided Math?
Most math programs are still set up in very traditional, teacher-centered constructs. In the name of “offering support,” some even provide scripts! This is typically followed by a barrage of worksheets. Then quiz tomorrow. Spiral review. Repeat.
Perhaps the monotony would be worthwhile if we all become mathematically literate adults, but this does not seem to be the case. As the National Center for Education Statistics keeps confirming in surveys conducted since the 1980’s, most Americans’ math skills remain lacking:
It’s time to look outside the box of traditional math education in order to foster life-long mathematical illiteracy!
One day, while complaining to another teacher about how I’d started hating the sound of my own voice, she introduced me to guided math. What I found most intriguing:
The use of math “stations,” even for older students
The possibility of teaching lessons to small groups (4-8 students at a time)
Easier access to limited math manipulatives
More time for individual students to receive what they need most, whether it’s practice, instruction, or extension projects.
I started literally the next day.
And while it took longer than that to refine my approach, the beauty of guided math is you can easily adapt your school’s math program to its structure.
Time Needed: 1 to 1 ½ hours block
Warm Up (first 5-15 minutes): Number Talks were one of my favorite ways to warm up (see this 3-page pdf for more details). At the end of warm-up time, write or project on the board any materials students may need to bring to each station.
Stations time (45-60 minutes): Students either rotate among or choose stations.
Wrap Up (last 5-15 minutes): Allow students to share any mathematical discoveries they noticed.
Mini-lesson: This becomes a much more flexible idea than simply delivering lessons to the whole class. Some options:
The teacher works with small groups with math manipulatives, individual whiteboards, or other resources that are difficult to share/manage in larger groups.
Set up a computer with a video on the concept of the day from free video databases like LearnZillion or Khan Academy. See a fantastic example of how a 4th grade colleague of mine uses her classroom blog to direct students to the video she selects (they have the additional convenience of checking out a mobile lab for the entire class during math). The video option can be especially helpful on days that you need to have one-on-one math conferences with students.
Practice: Students try out concepts learned within the unit or the lesson of the day.
Reflection: Students record their math thinking and processes in journals.
Choose a Structure: Rotations vs. Choice
Rotations: Divide your students into 3-5 groups (mixed or leveled based on benchmarks, quizzes, or daily formative assessments). Take the length of math block time, subtract 10 minutes for whole-class time at the beginning and end.
Choice: Right after Warm-Up, take a status of the class, asking your students which 1-2 stations they will be working on that day and why. You may choose to require all students to select the mini-lesson and/or practice stations each day before choice time, but that depends on your students’ needs!
Don’t be afraid to try out both options a couple of times! Ask students to notice successes and issues, and to be ready to report back during the wrap up or weekly class meeting. Give them the opportunity to solve problems, and they will surprise you!
Model, Model, Model!
Practice examples and non-examples of every station as a whole class.
Issue: Students become off-task at the game and/or fluency station
Possible Solution: Ask for parents to volunteer during guided math, either to help check off, help students with their practice, or even to bring a math game to share with groups! You can also simply consider the location of your stations.
Issue: Students don’t get to every station every day
Possible Solution: That’s ok! If you’re doing rotations, just cut out one or two of the stations you’re using. If you’re doing choice time, just have them choose 1 station a day beyond the mini-lesson and practice.
Issue: Instruction time not long enough
Possible Solution: If you don’t find a LearnZillion or Khan Academy video you like, make a video of yourself teaching the concept! Not only can it help you say things more succinctly and briefly, but your students can individually pause, rewind, and rewatch as many times as they need to.
Issue: Students don’t have enough time to finish worksheets in the practice portion.
Possible Solution: Become a more careful curator of your resources–sure, your manual assigns 38 problems to practice adding fractions, but is that really what your students need most today? Or do they really just need to practice the 4 problems that involve mixed numbers? Or maybe, they need you to design a challenge activity that gets them thinking more about the concepts behind fractions. Never assume that the math textbook knows more about your students’ daily needs than you do!
Any other questions, tips, or experiences? We’d love to hear about them in the comments below!