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"The ultimate goals of mathematics instruction are students understanding the material presented, applying the skills, and recalling the concepts in the future. There's little benefit in students recalling a formula or procedure to prepare for an assessment tomorrow only to forget the core concept by next week. It's imperative for teachers to focus on making sure that the students understand the material and not just memorize the procedures. Here are six ways to teach for understanding in the mathematics classroom:"

"In order for students to be efficient in computational fluency, they must develop mental math skills and recall math facts automatically. Recall is a developmental process that improves computational fluency by developing efficiency, accuracy, and flexibility with numbers. The focus of instruction should be on thinking and on building number relationships. Facts become automatic for students through repeated exposure and practice. When facts are automatic, students are no longer using inefficient means, such as counting."

"In school, it was those who could recall the facts, and particularly those who could recall them quickly.  If you could memorize your multiplication tables you were quickly labelled as "smart".  Smart was a product of a system based on sorting - some kids were smart, and the other kids were . . . well, we didn't really call them anything aloud, but the implication was that they were less than smart. And in the traditional school smart hierarchy - the matching of provinces and capital cities along with the ability to memorize weekly spelling words was the apex of smartness. Of course, the last 20 years have moved us away from a world of knowledge scarcity to knowledge abundance; now, all manner of information is available to everyone. For better or worse, we no longer look to our political and intellectual leaders for their all-knowing guidance, we quickly check what they have said with what we read on Wikipedia, Web Doctor MD or other online information available to us."

"From ambitious entrepreneurs to casual meet-up participants, women are proactive about engaging with STEM and with other women in their field. Besides their shared affinity with science, tech, engineering, and math, they often find that they have another thing in common. Many of these women can recall a time when they walked into a classroom or office, looked around, and noticed that they were sharing the space with an overwhelming majority of men. This was the experience of Nicole Messier, a former student of aerospace engineering, who recalls that she had sparse female role models, teachers, or peers in her undergraduate program. Instead of following a traditional aerospace career, Messier instead decided to help change STEM's evident gender imbalance. She is now the CEO of blink blink, a purveyor of creative circuit kits that help beginners get their feet wet with simple electronics, arts & crafts with embedded circuitry, and wearable fashion technology. Not only did she co-found the company, she also collaboratively designed the kits with the very girls that they're made for."

"Like many kids of my generation growing up in India, I was an avid reader of Enid Blyton's novels. Many of her books were written as a series ("The Famous Five," "The Secret Seven" and "Five Find-Outers") and I recall wondering if the lives of characters overlapped in any way. Did a character from one series ever run into one from another, for example? I recall wondering the same thing in later years about P.G. Wodehouse's Blandings Castle and Jeeves series. Today, in a world where communities real and imagined are digitally connected via platforms like Facebook, Instagram and Tumblr, we can reframe that question in terms of those common nodes (or friends) in those characters' social networks. As it turns out, network theory as an analytic technique, or what I'd call computational literary analysis, is not just a bona fide research endeavor. It's also a great example of how computational thinking (CT) is truly a cross-disciplinary skill that can be weaved to enrich learning in any subject (not just math and science, as is sometimes the assumption). In an earlier article on computational thinking, I offered teasers of how CT could be integrated into language arts and social studies, in addition to math and science. Here's a detailed treatment of one of those examples, drawn from the work of Franco Moretti's group on "Computational Criticism," which is part of the broader Digital Humanities initiative at Stanford. (See this New York Times profile for more on the work of this group)."

"Higher-level thinking has been a core value of educators for decades. We learned about it in college. We hear about it in PD. We're even evaluated on whether we're cultivating it in our classrooms: Charlotte Danielson's Framework for Teaching, a widely used instrument to measure teacher effectiveness, describes a distinguished teacher as one whose "lesson activities require high-level student thinking" (Domain 3, Component 3c). All that aside, most teachers would say they want their students to be thinking on higher levels, that if our teaching kept students at the lowest level of Bloom's Taxonomy-simply recalling information-we wouldn't be doing a very good job as teachers. And yet, when it's time to plan the learning experiences that would have our students operating on higher levels, some of us come up short. We may not have a huge arsenal of ready-to-use, high-level tasks to give our students. Instead, we often default to having students identify and define terms, label things, or answer basic recall questions. It's what we know. And we have so much content to cover, many of us might feel that there really isn't time for the higher-level stuff anyway. If this sounds anything like you, I have a suggestion: Try a curation assignment."

"My brother George recently wrote the post "Denying Our World" where he recalls a compelling narrative that causes him to reflect upon what it means to live 'online' and our associated imperative as educators to teach to this reality. "

"Since the beginning of September we have been trying to maximise the use of 1:1 iPods in year 6 in all areas of curriculum. The potential of enhancing teaching and learning in mathematics through the use of this technology has been particularly interesting. We have been developing the creative use of a range of Apps to support progress, engage children and add relevance to maths teaching with positive outcomes (10 Practical ways to use Apps in Maths) We have also explored a wide range of maths specific Apps which have helped pupils mainly in the areas of number fact and tables recall. (Apps for Maths) Recently we have extended the use of the iPods to allow them to support independent learning, and play a central role in effective formative assessment."

"Patrick Larkin, Assistant Superintendent for Learning, Burlington Public School District, Burlington, Ma, recalls a question someone posed to him: "Is the tablet an education solution, or an education problem?" His answer, and common sense approach to changing the school and learning environment is a professional development lesson for all. Enjoy my interview with Patrick Larkin, and learn why he's become one of the most respected tablet education voices today-and for the future."

"As first graders are learning about the butterfly life cycle, we wanted to stay away from usual activities such as coloring in a pre-printed coloring page. INSTEAD of such an activity (created by others) and a quiz about recalling the different stages of the life cycle as assessment, we decided to have students built on their knowledge and fluency of creating a collage and CREATING a visual of their learning. The digital visual was to become an artifact for their student portfolio."

"Can you recall a time when you were so engaged in learning that you became unaware of your surroundings, that each step of the process energized you to pursue your goal further, and the learning became its own reward? Recently, I was working with a gathering of administrators from my county, contemplating the puzzle of student engagement. Most of the small groups in the room organized their exploration around the premise that on-task behavior is a necessary but insufficient step toward engagement. The T-charts they created focused on how an administrator observing a class might recognize subtle differences."

"This week we've been reading a favourite winter time story! The children love the adventures the Gingerbread Man has as he runs away from the different characters in the story. They are always so sad at the end when he is eaten by the fox!     Today I challenged them to retell the story and help the Gingerbread Man outsmart the bakers, cow, goat, cat and fox by using a coding game. This activity helped the children to strengthen their retell as they recalled the story and sequenced the events of the story, and encouraged them to practise their coding skills, specifically oral language related to directionality and number.   We used a small stuffed Gingerbread Man, photos of the different characters from the text, our coding board (sensory table lid with tape), and directional coding cards. "

"Root looks like a smoke detector but is actually a sophisticated robot. A magnetic surface, wheels, and an impressive arsenal of sensors allow it to navigate a classroom white board. But Root isn't actually programmed to do anything. Its tasks and functionality hinge on a child's imagination. To operate - Root needs instructions, a line of code. Zivthan Dubrovsky of Harvard's Wyss Institute recalls testing out Root with kids for the first time. He asked them this: "Can you make a text based java script line follower? They go 'no that's hard, can't do that', but we can put level one in front of them and they can do it in minutes." Level one introduces kids to principles of programming using an interface of simple instruction and pictures. As they become more adept, they jump to levels 2 and 3, at which point writing computer code becomes second nature, according to Dubrovsky."

"When in Boston, I taught next to a Geometry teacher who would later go on to become Teacher of the Year. Lining the walls of his high school Geometry classroom, from floor to ceiling, were vocabulary words with drawings and examples. At the time I thought it was a bit extreme. I mean, aren't these kids in high school? Over the years since, I have come to realize just how important word walls are and that he was absolutely right in putting it all out there like that. If you think about it, there's a measly 5 year age difference between a 4th grader and a 9th grader. Yes, kids do grow up quick, but what is it about those 5 short years that suddenly allows students to remember and recall everything and no longer need visual reminders?"

"Every teacher I've worked with over the last five years recalls two kinds of digital experiences with students. The first I think of as digital native moments, when a student uses a piece of technology with almost eerie intuitiveness. As digital natives, today's teens have grown up with these tools and have assimilated their logic. Young people just seem to understand when to click and drag or copy and paste, and how to move, merge and mix digital elements. The second I call digital naiveté moments, when a student trusts a source of information that is obviously unreliable. Even though they know how easy it is to create and distribute information online, many young people believe -- sometimes passionately -- the most dubious rumors, tempting hoaxes (including convincingly staged encounters designed to look raw and unplanned) and implausible theories. "

"There has been plenty of hand-wringing in recent years about the "overscheduled child." With after-school hours increasingly dominated by piano lessons, soccer practice, and countless other planned activities, many of us have a nagging sense that kids are missing out on something important if they have no time for unstructured play. New research from Germany suggests these fears are justified. It finds people who recall having plenty of free time during childhood enjoy high levels of social success as adults."

"Today while I was browsing through my Twitter feeds I came across this fabulous Bloom's Taxonomy wheel of apps shared by Anthony. If you still recall, some previous versions of this wheel have already been featured here in Bloom's Taxonomy for Teachers section . As you can see, the wheel outlines a wide variety of verbs and activities related to each thinking level of Blooms taxonomy coupled with iPad apps that go with it. These apps are supposed to help teachers and students better cultivate these different thinking levels in their use of iPad apps. And because the the visual is not hyperlinked, I went ahead and provided the links for each of these apps in the lists below. Enjoy"

"On a recent flight to Buffalo, on a recommendation from Amy Burvall (@amyburvall) I started reading Austin Kleon's Show Your Work I don't remember the tweet exactly, but I recall that Amy shared something along the lines of..."This is the best book about education that isn't about education." I didn't have much choice but to read. Halfway into the book I stopped reading, popped open my iPad and had to start writing this post. Grab of copy of Show Your Work and you will quickly understand wh"

"Questioning is a fundamental element of pedagogy, one you could read endlessly around, but the reality is using questioning to challenge and engage all learners is demanding and potentially problematic to get right. Recently I've been working with a team of teachers, shaping our CPD model in preparation for the new academic year. Engaging in dialogue around teaching and learning with colleagues is always a pleasure and extremely informative, and one aspect continually crops up; deep, challenging and engaging questioning. Firstly, I think it's crucial to outline what we are trying to achieve when we think about the purpose of questioning, for me it includes the following: Allowing students to develop a fuller understanding of a concept because they have tried to explain it themselves To easily recall existing knowledge To be able to link the ideas in the lesson with existing knowledge To tackle problems at a deep level and be able to extend their thinking To engage easily with a task because they are clear about what is expected To develop independence in the way they learn and think"

The time between the end of one school year and the beginning of the next is a season for reflection and renewal. It's important to celebrate student successes and recall challenging times when things just didn't go as planned. This season between school years is an opportunity to plan for new and modified instructional approaches to increase learner success. What follows are suggested readings and resources for developing and deepening your practice to meet the diverse needs of all learners. The readings are grouped from easy starts to deeper complex implementation. Like Steve Jobs, feel free to choose what captures your passion.