Math Archives

Subitizing Is An Important Math Skill

October 18, 2020 Brad Hoffman

While “subitizing” may not be a well-known word, it is certainly an important mathematical skill. Subitizing is the ability to instantly recognize the number of objects without actually counting them. Much like the importance of being able to calculate estimates, subitizing is something that comes up in the everyday lives of students. One example is counting the dots on the faces of dice: when you roll a six, chances are you don’t actually count out each of the dots. Rather, you have come to recognize the pattern of three rows of two as being equal to six.

According to The National Council of Teachers and Mathematics, one key part of effective subitizing is developing pattern recognition. Moreover, this mathematical skill allows students to gain a grasp of numbers and advance to higher levels of addition. As an activity, the NCTM suggests taking “snapshots”. Taking snapshots means showing an image of a collection of objects to the student for just a second, then asking them to take a mental “snapshot” in order to identify how many objects are included in the image.

As author of Subitizing: What Is It? Why Teach It? Douglas Clements says, “Students can use pattern recognition to discover essential properties of numbers, such as conservation and compensation. They can develop such capabilities as unitizing, counting on, and composing and decomposing numbers, as well as their understanding of arithmetic and place value—all valuable components of number sense.”

Be sure to check out this short video for more information:

Amazing Math: Subitize Me! Trailer 1

Watch this video on YouTube

Here are a six great apps for strengthening students’ subitizing and group counting skills:

By Brad Hoffman, M.S.Ed.
Board Certified Educational Planner and Learning Specialist
My Learning Springboard

Schooling a Math Prodigy: A Memoir

October 6, 2020 Brad Hoffman

$math prodigy$ At elementary school age, many thought I was a math prodigy. Several schools and countries later, I wasn’t so sure, but my dormant math talent mysteriously re-awakened at around adolescence, a time of re-discovery and of finding your place in the world.

My mathematics development worked out fine from there except that I was self-taught from eighth to twelfth grade and there were no internet resources in those days so I studied from my father’s engineering textbooks which meant that when I reached college, I knew how to solve partial differential equations but had a limited grasp of linear algebra. I was determined that if I ever had a child with the same mathematical gift, I would nurture that spark at an early age and keep it glowing.

It was a challenge. I became the “parent from hell” to my son’s second and third grade teachers. Having been a Middle and High School math teacher I was shocked at how fearful of mathematics many elementary school teachers seemed to be. My child and others were bored with the repetitive activities they were expected to perform.

Teachers seemed to aim at just below the middle of the class so that most felt they were geniuses, whereas in fact they were simply not being challenged. As a Middle School teacher, I remember once being instructed to give one of three grades on my report – Excellent, Good and Above Average.

Self-esteem trumped all other considerations – but at what cost?

Conversations with teachers became unproductive – I was accused of teaching my son too much at home, thereby making school boring. My son was also instructed to help other students for the whole of third grade – a worthwhile endeavor up to a point, but at no time was there any desire to challenge him. Class size was one of many problems, so when a private school insisted they would challenge him no matter what his level, we jumped at the opportunity.

At first, there wasn’t a noticeable change – he was simply given extra worksheets, but there was a recognition that he needed more stimulus than most others in the class. He was given more advanced textbooks, which he would work on when he finished his classwork, but it felt more like “punishment” since regular schoolwork was time-consuming even if it was easy. The break came halfway through fifth grade when his teacher realized that we were running out of materials and offered to put him on the online program (ALEKS) that the Middle and High school was using.

He still had to do regular classwork, but he was more energized now since there was nothing holding him back. He could learn by himself, without waiting for me to have time to teach a new topic. The program gave him multiple ways of tackling problems and he looked at all of them before deciding which one he liked best, putting him more in charge of his studies. There were frequent tests, at the end of which he would have to re-do topics he hadn’t covered well enough, while simultaneously covering new material – a fruitful balance of review and new learning. When he started Middle School, he was allowed to skip math class – the other option was to join tenth grade math, but it was socially too daunting.

Instead, he finished his non-math homework in those “free” lessons, and did math at home with me. I felt it was a huge decision to hand over this responsibility to me, and I always worried about how I could “prove” to the administration that I was being responsible. One way, I thought, was to try standardized testing. With no preparation, my son took his SAT I exam, not having much clue about the English portion but scoring 720 on the math – enough to get him into the prestigious SET program run by CTY. For the writing portion, I told him to write about chess, no matter what the topic.

The question was “Do you think equality will ever be achieved?” and he started with “On the chess board, the pieces are unequal but in certain situations, a weak piece can take on the same value as a supposedly stronger piece…”

K had actually completed Pre-calculus by the time he had taken the SATs in sixth grade and then completed all the material for AP Calculus AB and then AP Calculus BC over the summer between sixth and seventh Grades – the Khan lectures were a valuable resource. He scored 800 in the SAT II Math 2 exam at the beginning of seventh grade and scored a 5 in the AP Calculus BC exam at the end of seventh grade.

Exam results and recommendations got him into CUNY (Hunter College) where he has started Calculus III at the beginning of eighth grade, going to college two evenings a week for college credit while still a full-time Middle School student. Is this a bridge too far? I’m not sure, but college courses were the logical outcome of his rapid ascent through the curriculum.

Did we cover things too quickly? Perhaps. There are gaps (e.g. geometry) but we have enough time to make mistakes and repair them before he becomes a full-time college student. The important lesson was to adapt our approach and give our son the tools he needed to thrive. To give our son the opportunity to do his best by adapting our approach as need be.

By Kovan Pillai, Private Tutor

Classroom Management Strategies for Distance Learning Success

September 20, 2020 Brad Hoffman

While a lot of the plans for this school year are changing from day to day, one certainty is that many students will be distance learning or experiencing a hybrid model of education. Naturally, families are concerned about how to support their learners. While parents often feel grounded in subject areas like math, reading, or history, or at least have good enough research skills to find helpful instructional videos online, many families feel out of their league when it comes to helping children strengthen their executive function skills, particularly in subjects that posed a challenge during in-person school terms. Quite simply, executive function skills are the skills we use to plan, organize, start, stay focused on, and complete a task. They are essential in every educational subject. The good news is that executive function skills are the same skills that adults employ to be successful in their lives and careers — which means that most parents have the aptitude to support their children as they develop these skills. And in fact, every student benefits from support with executive function skills during distance learning. One of the best ways of helping students strengthen executive function skills at home is by employing successful classroom management strategies.

Classroom management strategies are the procedures and guidelines that skillful teachers put in place in order to make instruction run smoothly; they are quite similar to executive function skills, but for a group of people instead of with one student in mind. Good classroom management systems take into account students’ needs in terms of learning space, resources, visual stimulation, bodily movement, time management, and self-monitoring. Because your child’s home learning space functions as a classroom space, classroom management considerations can help you and your child establish an optimal home learning setup.

The best way to identify which classroom management strategies your student needs help with is to have a conversation with them. Approach the initial discussion as a brainstorming session where all suggestions will be considered. Be open to listening to your student, and be willing to consider ideas they generate. If you are not sure whether an idea is feasible, agree to try it out for a trial period. It is essential for students to be deeply involved in this process, as it helps them build their executive function skills and increases their sense of agency and their investment in their learning.

Before School: Classroom Management Strategies

A good way to start the conversation is with the “Before.” These are the things that need to happen before the student starts school each morning. List the routines you can establish as a family to start the day successfully. What time must they be up to allow enough time for breakfast, getting dressed, getting ready? How can you build in some physical movement to help your child wake up and to give a sense of transition — is it possible to take a short, brisk walk together before class? If a walk is not possible, can you turn on some music and dance together before the school day begins? Then, make plans for what happens during learning breaks. If possible, try to eat lunch together; if not, let them know what the plan will be for lunch. Make sure your student knows how to get your attention if they need you during class. To the greatest extent possible, keep these routines and guidelines consistent from day to day.

During School: Classroom Management Strategies

Next, begin to talk about the “During.” Ask your child: What worked best for distance learning in the spring? What was really hard or could be improved upon? Are there things they wish would have happened differently?

As you plan for this year’s learning, one thing to discuss with your child is their spatial needs. If they often felt distracted with previous distance learning, take a good look at their workspace. Some older students might prefer working solo in their bedrooms, while some younger students might find it lonely. Also, some families have limited space, and having a quiet, private room with a door might not be possible. As a family, discuss what the options could be. Talk about the places where your student might learn best: At a desk? On the floor? In a comfortable chair? With younger children, setting up a yoga mat or beach towel allows them to move around during class within a defined boundary and camera range. Once you have established a learning setup, figure out how to optimize that setup for learning. If a student will be using their bedroom, discuss why it is important to make the bed each morning and to turn off and put away distractions like toys, TVs, and phones. Make sure the space is comfortable and sends a message that learning is happening there. Once you settle on a workspace, create guidelines for how other family members should use or avoid that room when the student is working.

Next, brainstorm the resources and supplies your student will need to have close by. If possible, consider a hard wired setup for your student’s computer in case the household WiFi lags, and get your student a headset with a microphone. Additionally, although your student is attending classes online, do not overlook the importance of physical supplies. Educational research demonstrates that most students learn better when they take notes by hand rather than when they type them. Maintaining a binder or notebook for each class, as they would during “regular” school times, helps many students stay grounded in the physical world and makes distance learning feel more “real.” Have your student help you choose supplies — pencils, pens, highlighters, crayons or colored pencils, maybe some stickers, the fun things they would buy in a normal year. We also recommend a small whiteboard with dry-erase markers, print copies of their books, and a water bottle to stay hydrated. A fancy custom storage system is not needed; a shoebox can hold the supplies, and your student can decorate it themselves if they’d like.

Visual needs are an especially important consideration with distance learning. While physical classrooms are often full of colorful, helpful and inspirational posters, learning online means that your student’s eyes are already absorbing a lot of stimulation. Eye strain is not just a problem for adults when staring at a screen for a long period of time — and anything over 20 minutes is a long time! Encourage your child to purposefully blink their eyes while online. Make sure their physical workspace is well lit. Together, decide whether posting a picture on a far wall or having a window view would give them something to look at to take a moment from focusing on their screen. If a student will be sitting at a table or desk, consider setting up a white trifold board in front of their workspace to help them stay focused. Some students benefit from posters on their board, such as printouts of their daily schedule and of the alphabet or the periodic table of elements. Some students thrive on personalizing the board, as they would a locker door. Others benefit simply from having a clean sea of white as a respite from the lights and noise online.

Physical needs during lessons are another topic to discuss with your student. If it is hard for them to sit still during synchronistic teaching, consider what you do to stay focused when you are at work. Can you teach them how to stretch in their chair? Can they drum their fingers on a pillow? Can they keep a fidget toy nearby that is soothing, but not distracting? Could you place a piece of stick-on velcro on the bottom of their desk or chair to run their fingers back and forth on? Can they rub their feet on a small textured mat? Would sitting on an exercise ball be better than sitting in a chair? Would a standing desk option by helpful?

Physical needs during breaks go hand in hand with physical needs during lessons. Make a list of physical activities and the places in your home where the activities can happen during scheduled breaks. Practice a few yoga poses until the student can do them independently; a few minutes in child’s pose helps most students quickly recenter. If one of your student’s scheduled breaks aligns with a time when you also can take a break, do a mini workout together. For some students, a few minutes alone listening to music may help them unwind. These activities are helpful not just during instructional breaks but during students’ independent work time, as they help the student break big blocks of time into manageable segments. Because students already are spending so much time online, encourage them to avoid break time activities that require a screen.

Just as physical note taking can help students stay focused and rooted in the moment, so can self-monitoring their learning. If they are listening to prerecorded lessons, encourage them to develop a habit of stopping the video occasionally, thinking about what they just heard, and stating it out loud and/or summarizing it in their notes. Younger children can teach what they are learning to a favorite stuffed toy sitting nearby. If a student struggles to recall important information or summarize the lesson effectively, they might need more support learning that particular concept.

After School: Classroom Management Strategies

After discussing the “Before” and the “During,” it is time to discuss the “After.” No matter where the workspace is, decide how your child will “close shop” at the end of the day. Where will materials be stored? Where will their device be charged? What transitional routines will help them to change the space from a learning space back to a home space?

Once you have used classroom management strategies to establish a workspace and some routines and guidelines, set a schedule for checking in with your student. Some students need a daily debriefing, while for others, every few days will work. Ask them open-ended questions and give them time to form their thoughts. Some questions might be:

What is working well in your workspace?
What changes do you think you need to make?
Are you able to manage the LMS (Learning Management System, i.e. Google Classroom, Canvas) the teacher is using?
Is there anything you want to ask your teacher or share with your teacher? Do you know how to reach the teacher? Have you reached out to the teacher? Have they responded?
Are you able to pay attention during instruction? What might make it easier to pay attention?
Do you need any new supplies?
What did you learn in reading? In math? In science?

Continue to observe what classroom management strategies are working for your student and which ones are not. Be sure to give them specific feedback when you see them using a strategy that is successful. Reach out to their teacher, or help them reach out to the teacher themselves, if you see them struggling. Be as specific as possible when describing your concern. Be open to the teacher’s suggestions and bring them to your next family brainstorming session. Post the day/time for that session on a family calendar where the student can find it easily. The important message your child needs to hear is that they can grow their executive function skills by trying new strategies and by reflecting on what works for them as a learner. With these improved skills, they will find more success in school not just this year but in the challenges that await them farther into the future.

Written by Maggie Sisti, M.Ed. and National Board Certified Teacher, who is a classroom teacher in a large, high performing school district in the Maryland suburbs of Washington, DC. She has been supporting her own four kids and 30+ years of students with developing and enhancing executive function skills inside and outside of the classroom.

Does My Child Need An IEP?

September 6, 2020 Brad Hoffman

This is a question that doesn’t always have a quick answer. An Individualized Education Plan (IEP) provides specialized instruction so that a student can access the general education curriculum. Prior to beginning the IEP process a number of things need to be considered.

If your school follows a standards-based curriculum, how is your child progressing? Is your child meeting the standards for their grade level with minimal support during and after the school day? If so, it is likely that beginning the IEP process is not warranted.
Is your child accessing the general education instruction in their classroom with accommodations in place? Is he/she meeting standards with these accommodations in place? If so, then consideration of a 504 Plan is worthwhile.
If your child is struggling academically, have you and your child’s classroom teacher(s) identified the specific areas of concern? Have concrete instructional strategies and interventions that can be implemented in the classroom and at home over a set period of time been identified (Response to Intervention [RTI])? What was the outcome? Did your child make the progress expected? If so, continue to identify effective strategies and implement them at school and at home. If your child did not make progress toward or achieve their goals, despite excellent instruction and on-target strategies, it may be time to begin the IEP process.

The Individualized Education Plan (IEP) Process

Prior to developing an IEP the student must be identified as having an educational disability. Parents are part of the team that addresses the question “does this student have an educational disability?” Other members of the team are usually comprised of a school-system psychologist, the child’s teacher(s), a special educator and often a school administrator. The team determines if they suspect the child has an educational disability and if further assessment is needed.

Generally, the team identifies questions that should be answered as part of the assessment process. Common questions might be:

What is the current level of STUDENT’S intellectual functioning?
What is the current level of STUDENT’S academic functioning?
How does STUDENT’S current level of intellectual and academic functioning impact their ability to access the general education curriculum without specialized instruction?

If the assessments and answers to these questions indicated that an educational disability is present, then the disability will be identified and the next step will be to develop an IEP. The IEP includes the current intellectual functioning and academic data. This information drives the identification of the goals and objectives which make up the heart of the document. The intensity of the support/intervention needed to meet the goals and objectives will determine the amount of service a student requires. This can range from 5 minutes a day to a full-day of specialized instruction.

Parents are often concerned about the stigma of their child receiving special education instruction. However, depending upon the structure of your child’s school and program, this intervention can be delivered seamlessly. As a special education teacher, administrator and consultant, I often ask parents, “If your child had significant allergies and they impacted their ability to do well in school, wouldn’t you intervene? If your child needed glasses, wouldn’t you provide them?” Almost everyone responds, “Absolutely.” This same response should be strongly considered if your child legitimately qualifies for special education support.

How to Avoid the Summer Slide

May 10, 2020 Brad Hoffman

Summer is fast approaching and many parents have already planned activities and camps for their children, or they’re in the process of planning or recalibrating. We know that summer is the ultimate freedom, the time for exploration and play, the time to finally unwind from the busy school year. So how do you maintain your child’s academic skills while enjoying the freedom and playfulness of summer vacation? A critical part of summer planning relates to academics and keeping students’ skills not only intact, but also progressing. Each year, we observe that some of our own students slide back several weeks or months when summer vacation does not include any academic practice. While many students can recover fairly quickly during the fall, this summer slide is an unnecessary setback that can negatively impact a student’s confidence and make going back to school more difficult.

When you stop and think about it, the contrast between the structure of the school year and the liberation of summer is profound. During the school year, students are asked to sustain attention for long periods of time, to read critically, to write reflectively, to problem solve, and to synthesize information. The summer is typically the yin to the school year yang.

But a balance can be achieved to keep those learning muscles strong while still allowing for some rest and recovery. We view learning as a marathon, not a sprint, and the current school year schedule makes that tricky. With proper planning and collaboration, parents can help their children to maintain a successful position without creating a household battle.

So how do you do it? Each situation, each learner’s response to instruction, and each teacher is unique. For summer success, especially this year, we encourage parents to develop a strategic plan in partnership with their school, child, and, possibly, private tutor. From camp to virtual field trips to kitchen-lab science, there are tremendous enrichment and experiential learning opportunities available.

During the summer, students also need to practice sustaining attention, synthesizing information, and problem-solving in more structured ways to avoid cognitive atrophy and allow for a fairly seamless transition back to school. Here are 5 tips for planning summer learning. The most important thing is to set yourself up for success. Choose from the suggestions below, commit to a schedule and hold strong. Follow your child’s lead, but insist on a minimum of 15 minutes for each activity to maintain stamina. But most importantly, have fun!

1) Talk to your child’s teacher by early June. Prepare a friendly email or request a brief conference asking for recommendations of skills or concepts to work on during the summer. You might refer back to report card comments to confirm if certain skills were mastered. Also, confirm if there is any required schoolwork for the summer. Many schools ask students to read a certain number of books, to write book reports, or to complete math practice work over the summer. Book lists or summer packets may be available.

2) Talk with your kids about their learning goals. Once you’ve gathered more information from the school, share it with your children and talk with them about their summer interests. Review your family calendar and share the options for summer activities. Make your kids part of the process.

3) Read daily. It takes discipline, but all students need to read daily for a minimum of 20 minutes. Students also need to be engaged in conversations about the text. It is incredibly complex to develop proficiency with summarizing, identifying the main idea, determining an author’s message, or explaining the theme. Students need tremendous practice and modeling. For independent reading, should have “just right” books at an appropriately challenging reading level. Many families have the most success when this reading time is scheduled first thing in the morning or as an afternoon “siesta” period.

4) Practice writing and typing (even drawing helps). There are lots of ways to practice writing over the summer that are meaningful and fun. Our students often like to keep journals for writing friendly letters to their parents, family members, or tutors about their reading, and they love getting a written response in return. Students also like to write stories and publish their own books. A great site to use is www.blurb.com, and summer journaling with a tutor can be managed locally or from a distance via online collaborative tools. Typing is also a real-life skill that every student should master at an early age. Software is available to help students develop proficient typing skills.

5) Practice math. Set aside at least 3-4 times a week for math practice. Math practice should include problem-solving, mastering basic facts, and building number sense. Workbooks, games, and websites are readily available in addition to customized private programs and math camps. IXL.com has practice activities for every grade level aligned with state standards.

We wish everyone a wonderful and well-deserved summer vacation and a smooth transition back to school in the fall.