Math Archives

Being The Teacher Is a Powerful Learning Strategy

August 19, 2021 Brad Hoffman

OK…as a parent, teacher, coach, or private tutor….it is our job to teach, right? Well, yes, but not always. One of the most powerful teaching tools we have is giving our students the “job” of being the teacher and teaching us. There are several specific reasons for this:

First, often the role of “purveyor of knowledge” is very appealing to kids. I have watched students of all types rise to the occasion of being the teacher. It seems especially appealing if they are asked to teach one or two younger children, or if they are asked to teach an adult!

Second, I have found that I cannot teach anything I don’t truly understand. In the process of teaching, I often uncover any slightly murky parts of my own understanding. This leads me to more thinking and better understanding of the topic. For students who take on the role of being the teacher, the same often holds true. As they try to teach me, they discover what they truly DO know, as well as what is still unclear. Finding out what questions they still need to ask makes them better learners.

Third, we learn a great deal about our students’ depth of understanding by truly listening to what they “teach” us. (Note: truly listening can be a challenge for those of us who are bursting with excitement to teach or who are up against a time crunch or any number of other stressors, but listening is essential.) In the role of “learner” we can appropriately ask questions which will clarify what they really know. We can also determine whether their knowledge is rote (memorization of formulas, timelines, etc.) or actual conceptual understanding of ideas and processes. In some cases, we can guide deeper understanding through creative questioning. Other times we simply uncover misunderstandings and/or shallow understanding that guides our next steps in instruction. Sometimes the discovery is not in the precise area of instruction but in a crucial related area! These “teachable moments” can be especially beneficial to our students.

One of my favorite collateral “wins” from giving students the powerful role of being the teacher is that they really do teach me new learning. My own, very legitimate, excitement about learning something new is contagious. It is valuable for students to realize that respected adults continue to learn, and that we are excited about learning, all the time. By being open to thinking about ideas in new and different ways, I am validating their thinking process and the control they have over their own learning; and I learn something new myself! Now that is powerful!

By Jean Snyder, Private Tutor

How Can We Help Our Students With Anxiety and ADHD?

April 15, 2021 Brad Hoffman

How often have you heard these statements?

“I just want to get this right.”

“If I can’t find the right word to use here the sentence won’t be perfect.”

“It’s so hard to put all the pieces together on paper.”

“What if I don’t get an A?”

“How do I compare with my peers?”

“Sometimes I worry so much that I can’t think about anything else.”

“There are so many things to keep track of, I’ll never get finished.”

“I never had trouble with school before, now I am, what’s wrong with me?”

“If I don’t turn it in, I won’t make any mistakes.”

“I’ll do it, just leave me alone about it.”

Anxiety and ADHD

Anxiety and ADHD — these diagnoses often come together and one may look like the other, but they are quite different when looked at closely and situationally. The Anxiety Disorders Association of America estimates that 50% of people diagnosed with ADHD also have an anxiety disorder, and anxiety in children and teens often overlaps with depression. Each of these diagnoses, separately and together, has a negative impact on executive functions, too. These overlaps can make it challenging to tease out the primary or driving issue, but what’s most important is identifying the contributing factors and strategically and systematically planning for intervention at home and at school to disrupt the negative loop.

ADHD

As diagnoses are refined, we find that students with ADHD (whether it’s inattentive type, hyperactive type, or combined type) often find it difficult, if not challenging, to focus and concentrate most of the time in a multiplicity of situations and locations. It’s often a struggle for a student with ADHD to follow multiple steps of a process, manage multiple deadlines, and shift gears without becoming overwhelmed. Students with ADHD can become anxious and panicked due to their worry about missing deadlines, not turning in work or struggling to pay attention in school as this is a pattern which has often been repeated with negative consequences and feelings of distress. Their challenges are generally not situational in nature, though it’s notable that children with ADHD can often focus extremely well, even hyperfocus, on a self selected and high interest task or activity.

Anxiety

Anxiety can be generalized as well; however, it is more often than not triggered by specific situations or events, which can cause students to find it difficult to concentrate and/or focus at that specific time. Students with anxiety may worry about their peer relationships (maybe also family relationships or the home situation) and how their peers view them both academically and socially. These students often become distracted by their worries, which can lead to the development of physical symptoms, such as stomachaches, rapid breathing, and trouble sleeping to name a few. Additionally, many of our students with an anxiety disorder also have perfectionist tendencies, and/or compulsive tendencies, which can render them unable to begin an assignment or assessment due to the anxiety about not being able to complete it successfully or perfectly.

With our very bright, persistent, and perfectionist students, it’s not uncommon to see some rigidity and performance anxiety. Attention and anxiety are closely related, and if a student is focused on “being right” or “not making mistakes” or “needing it to be perfect”, it can make it difficult to begin, or complete, a new and challenging task. Moreover, with these students who are used to “getting it” on the first try, being in a situation where they might not “get it right away” can feel incredibly humbling and frustrating, thus leading to avoidance and/or resistance and overwhelming anxiety. We often see these challenges with our students who are openly labeled “gifted and talented” at very young ages and then encounter certain academic challenges as they age up.

How do we help students manage their anxiety and ADHD?

First, we have to consider the specific factors, variables and triggers in each child’s situation. An intervention plan should be carefully developed that considers both the home and school environment as well as any other activities or transitions in a student’s week. Small changes can often have a big impact, so we like to look for those opportunities first. If there’s a longer history or more escalated concern, the value of a comprehensive evaluation with a skillful and thorough psychologist cannot be underestimated. Then we help families to translate that information into an actionable plan at home and at school.

Second, all of our students, and especially our students with anxiety and ADHD, benefit from consistency, predictability, and order. Having and practicing routines leads to mastery and ease with transitions and provides great comfort to children. Parents can establish morning routines, after school routines, evening routines, bedtime routines, and weekend routines. At the same time, we need to also help our children develop comfort with flexibility and change. For our children with high levels of anxiety, this flexibility can be extremely difficult, so we need to plan for practice opportunities in a safe space. Students with anxiety and ADHD also benefit from having clean, organized, and thoughtful spaces at home and at school. Organized spaces provide comfort and structure while modeling successful executive function skills for our students.

Third, all of our students, and especially our students with anxiety and ADHD, need parents and teachers to impose technology boundaries. Technology use is a major “time robber” and contributor to off-task behavior and anxiety in children and adolescents. Teaching our students how to responsibly use technology as a tool is critical to their academic and personal success. Achieving responsible use is a much more complicated discussion, but undoubtedly it starts with planning, discussing, and establishing expectations that are enforced with consistency, predictability, and transparency. To the extent that expectations and messages can be consistent between home and school is especially valuable, but it’s okay to have specific expectations at home that differ from school or from friends. Creating a “phone jail” during homework time and dinner time could be very useful for all members of the family. Consider “silent” times when all mobile devices are set to silent or “do not disturb.” Consider “screen free” times within the course of your family’s week. Parents and teachers need to model their own responsible use as well.

Fourth, we encourage families to consider a second sandbox when thinking about school placement. Boarding schools (and Junior Boarding Schools) — which come in many shapes and sizes — are often a better environment for our students with stress sensitivities triggered by city stimuli. New York City is incredibly fast-paced and energized, which is great for the adults who choose to live here, but it’s not always so great for our students. We’re fortunate to have such easy geographic access to so many campuses offering incredible academics, learning support, advisory, arts, sports, and diverse communities, which become extended families and neighborhoods. For students needing therapeutic school environments, we have tremendous resources in that category as well.

Faya, Laurie, and I help families everyday with educational problem solving, which includes school planning and school placement, developing and implementing intervention plans, refining (or overhauling) their weekly schedules, and partnering with their schools more effectively. Whenever possible, we work in close collaboration with psychologists, psychiatrists, pediatricians, occupational therapists, speech therapists, school leadership, school learning specialists, and classroom teachers to optimize resources and coordinate planning. A big challenge for families is knowing what to do at home to support learning and how best to partner with their child’s school. Without this close partnership between home and school, it’s very difficult to successfully mitigate challenges around anxiety and ADHD. If your child is struggling, we can help. Please contact us for more information.

These links may also be of interest:

Anxiety Disorders Association of America

Anxiety and Depression Association of America

NIH: Anxiety Disorders

CDC: ADHD

Child Mind Institute: Resources for Parents

Written by Brad Hoffman, M.S.Ed., Board Certified Educational Planner and Learning Specialist, Faya Hoffman, M.A., Board Certified Educational Planner and Learning Specialist, and Laurie Gross, M.A.T., Special Educator and Reading Specialist. Brad, Faya, and Laurie comprise the senior Leadership Team for My Learning Springboard.

Problem-Solving Strategies

January 26, 2021 Brad Hoffman

$Math-Manipulatives$ Problems! Problems! They come in all sizes and types. Some people claim that all of life is a series of problems to be solved, from the simplest (orange juice or milk for breakfast?), to much bigger (what occupation shall I pursue?), to the really big ones (how shall we achieve world peace?). When approaching a math problem, there are some important preliminary steps. First, it’s beneficial to know the standard math operations: addition, subtraction, multiplication, division, and their extended uses in working with common fractions, decimal fractions, time, money, and measurement. Having basic facts mastered and readily at hand is unquestionably helpful and necessary. But simply being able to calculate is not enough. Math is only truly useful when it is applied to solve a problem. For many students, these so-called “story problems” or “word problems” are where the trouble starts, but making effective use of universal problem-solving strategies will help.

Certainly, many students find that it is possible to solve a given word problem with minimal consideration of how to approach it. People have varying degrees of “math sense.” Some find most math problems mysterious. Some, however, can very easily see what to do to find solutions; it almost seems obvious to them. But even for students with strong “math sense,” there come those situations when they don’t intuitively know what to do. For all learners, the recognition of specific problem-solving strategies to solve math problems is useful. Thinking about our own thinking (aka metacognition) is important in developing flexibility so that we can see more than one way to solve a particular problem. Math journaling supports this thinking and development.

Below you will find a list of some very useful problem-solving strategies. One thing that is particularly beneficial about this set of strategies is that they are, in fact, universal. In other words, they will work regardless of the math program a student might be using. Whether it’s Singapore Math or Everyday Math or something else entirely, these problem-solving strategies can provide a clear path toward solutions. Interestingly, they can even extend to problem-solving outside the area of math! Becoming familiar with them and comfortable using them can be a big help to students as they wend their way through problems, be they less or more complex.

10 Problem-Solving Strategies

Make a model/Act out
Draw a diagram or picture
Look for a pattern
Make an organized list
Make a table
Guess & Check
Make it simpler
Work backwards
Use logical reasoning
Brainstorm

Here are some examples of problems and how to use these strategies.

Problem 1

“How many complete turns does the hour hand on a clock make in one day?”

From the list of problem-solving strategies above, “make a model or act it out” is an excellent choice for this problem. A student could use a model or a real analog clock and turn the hands and count. Distinguishing between the minute and the hour hand and recognizing that the clock only shows 12 of the 24 hours in a day lets the student see that the hour hand makes two complete turns. A physical clock that a student can actually turn provides an important concrete experience that may prove helpful for finding the solution.

Problem 2

“Using each of the digits 0, 1, 2, 3, 4 only once, make a two-digit number times a three-digit number multiplication problem with the greatest product.”

Students can “draw a diagram or picture” of an “empty” multiplication problem with a box for each digit. Consider which two digits give the largest product and put them in the highest place value spots. Then, if it’s not immediately evident to the student, use one of the other problem-solving strategies — “guess and check” — to place the remaining digits in the remaining spots. Check by multiplying the results to identify which is actually the largest (e.g. Is it 430 x 21 or 320 x 41?)

Problem 3

“How many even numbers are there between 201 and 351?”

In this instance, “look for a pattern” would be especially helpful from the list of problem-solving strategies. Either write all numbers from 201 through 351 and notice the pattern that there are 5 in every set of 10 numbers (e.g. 201-210), and then count how many sets of 10 numbers there are and multiply that by 5, or simply write one set of 10 numbers and identify the 5 in 10 pattern without writing out all of them. Either way is valid.

Problem 4

“You have two noses and three hats. How many different nose-hat disguises can you wear?”

For this problem, “make an organized list” from the problem-solving strategies listed above works well. The list will start with Hat A and match with each nose (2), then Hat B with each nose (2), then Hat C with each nose (2). This gives a total of 6 disguises.

Problem 5

“How many numbers between 10 and 30 give a remainder of 2 when divided by 3?” You could “make a table” to find the solution.


Number	Divided by 3	Remainder	2?
11	3	2	yes
12	4	0	no
13	4	1	no
14	4	2	yes
etc.

As the Table continues, a pattern becomes evident (“look for a pattern” — overlapping strategy!) in which every third number gives a remainder of 2. Count them for a solution.

Problem 6

“If 25 Glinks equal a Glonk, and 15 Glonks equal a Glooie, how many Glinks equal 2 Glooies?”

Please, “make it simpler”! That strategy is an especially good choice from the list of problem solving-strategies. Let’s look at a simpler, but similar, problem. It’s simpler because the numbers are smaller, and you could even draw a picture to prove it’s correct.

If 3 Glinks equal a Glonk. And 2 Glonks equal a Glooie. How many Glinks equal a Glooie? Multiply 3×2, which equals 6.

So, if 6 Glinks equal a Glooie, then how many Glinks equal 2 Glooies? Multiply 6×2, which equals 12. So, 12 Glinks equal 2 Glooies.

Now with the larger numbers:

If 25 Glinks equal a Glonk. And 15 Glonks equal a Glooie. How many Glinks equal a Glooie? Multiply 15×25, which equals 750. So, 750 Glinks equal a Glooie.

Then, how many Glinks equal 2 Glooies? Multiply 750×2, which equals 1500. So, there are 1500 Glinks in 2 Glooies.

It’s the same process, with bigger numbers! Much simpler!

Problem 7

“If I add 10 to my age and double it, I am 90. How old am I?”

From the list of problem-solving strategies, this problem begs for the student to “work backwards”. Simply un-double the 90 and subtract ten. 90 divided by 2 = 45 and 45-10=35. Voilà! The answer is 35 years old! Then reverse again to confirm that the answer is correct.

Problem 8

“Arrange these digits and symbols to make a true number sentence (equation.) 3,1,4,9,+,/,= (Note: the forward slash [/] signifies “divided by”.)

“Use logical reasoning” to realize that any order is possible, but a larger number needs to be divided by a smaller number with no remainder (9/3=3) Then 3+1=4, so the sentence 9/3+1=4 is the solution.

For the problems that seem absolutely impossible to solve, your best option is to “brainstorm”, and that’s on the above list of problem-solving strategies! Try various ideas; work with a partner; explore to see what might work; try everything you can think of! It’s amazing how good ideas will sometimes just pop into one’s head!

As a student works with these problem-solving strategies, it becomes clear that they often overlap (as in the “draw a picture” / “guess and check” example above, problem #2). Or a student becomes especially attached to a few particular strategies that often work well. Some problems seem to be especially suitable for a particular strategy, while others can be approached from several directions. Having the flexibility to move from one strategy to another helps avoid the serious “I’m STUCK!” situation. Also, using more than one strategy on the same problem allows students to check solutions more efficiently before moving on. Again, however, THINKING about how we are THINKING is very beneficial in developing skills in this area. We call this metacognition.

Solving word problems can be fun, like being a detective who has unusual insight. There are solutions! Enjoy finding them! And make effective use of problem-solving strategies!

By Jean Snyder and Brad Hoffman, Elementary Math Specialists

AP Exams and SAT Subject Tests

January 25, 2021 Brad Hoffman

While SAT and ACT scores demonstrate comparisons of a student’s skills in reading comprehension, English language, and math, the AP exams and, previously, the SAT subject tests, allow applicants to shine by testing subjects they love and often plan to pursue in college.

SAT Subject tests — which the College Board abruptly “retired” in January 2021 — were an important part of the admissions process, especially for the most competitive colleges. Now, students need to leverage the College Board’s AP exams to demonstrate subject mastery. However, AP exams can only be taken once annually in May and are administered at schools. Particularly over the past decade, many private day schools and boarding schools stopped offering AP classes because their faculty didn’t want to be boxed into a curriculum scope and sequence designed by College Board and argued by many secondary educators to be “a mile wide and an inch deep.” Instead, many rigorous independent schools have argued in favor of leveraging their advanced or honors classes for this testing, but that often leaves both content gaps and lacking test-taking skills that need to be supplemented with outside preparation. Moreover, many of these schools stopped coordinating and hosting these AP tests, and it’s not so easy to find another host school, which is an obstacle we’ll face this school year and next school year as educators navigate this testing change.

The TOP 6 AP Exams in 2020 were:

English Language and Composition

US History

English Literature and Composition

World History

US Government and Politics

Calculus AB

The TOP 3 SAT Subject Tests from 2018-2020 were:
Mathematics 2

Biology E/M

US History

Ideally, students initiate their planning and preparation for an AP exam at least 6 months before their anticipated exam date. The good news is that all of these tests are highly coachable, and there are scads of success stories among our tutors and students–the key is to start your test preparation early enough, and work with a specialist who is intimately familiar with the test(s). The best results come from a careful and strategic approach.

As a person who has experienced the admissions process year after year, I find that students who are interested in a particular subject, such as Physics, Chemistry, US History, or French, tend to score very well on AP tests, or, previously, SAT Subject tests because they study something they already like more deeply.

By Brad Hoffman, Board Certified Educational Planner, and Grant Bergland, Private Tutor

SSAT and ISEE Test Preparation: Why And For How Long?

January 5, 2021 Brad Hoffman

ISEE test preparation, why do it? How much time should be spent on it? THAT MUCH? Is the ISEE different from the SSAT? Those are really good questions.

The Independent School Entrance Exam (ISEE) and the Secondary School Admission Test (SSAT) are used for private school and boarding school admissions to grades 3-12 and skew way above grade level in what they assess, so just learning test-taking strategies is insufficient. Moreover, earning stanines of 7s, 8s, and 9s requires near perfect section scores. That’s why we recommend a nice long runway to prepare for the ISEE or SSAT. Most of our students are engaged in 1:1 programs meeting twice weekly–once with a literacy specialist and once with a math specialist–for 15-25+ weeks, which translates to 45 to 90 hours of instruction (plus homework). Yes, that much and here’s why.

At My Learning Springboard, we believe that ISEE test preparation, and most other preparation for standardized assessments, should be aligned with and extended from a curriculum framework. What we are doing should tie in closely with what your student is learning or will learn in his or her classroom. All learning is enhanced when it is connected with something that you already know and/or something you want to know. Preparing for an assessment such as the ISEE helps streamline that focus towards a multi-faceted endpoint; not only does your student solidify their foundational skills, but they extend them while also implementing targeted test-taking strategies. For this reason, we view ISEE test preparation as an opportunity to bolster middle and high school readiness. My Learning Springboard also offers specialized instruction for students with learning disabilities and/or heightened anxiety. These students also benefit from more time to learn, apply, and integrate testing content and strategies.

Over time we have found that the approach to ISEE test preparation that works best for families and students is one that allows for incremental improvements over time. We always start with diagnostic testing so that our recommendations are individualized, and we recommend this planning begin in January, February or March prior to the upcoming admissions season. We need this runway to effectively plan and implement both grade level and above grade level instruction in all areas of the test, including Verbal Reasoning (vocabulary development), Reading Comprehension, Quantitative Reasoning, Math Achievement, and 5-paragraph essay writing. ISEE preparation provides ample opportunity for enhancing executive function skills, such as planning with the end in mind, managing time more effectively, initiating and completing tasks, and controlling impulses and emotions.

Cramming for the ISEE is not effective. Taking the long-view for test preparation–making it a marathon, not a sprint–allows each student to adjust their pace based on numerous factors, such as diagnostic results, learning style, previous experiences, family and school commitments, and time for practice, practice, practice. Taking a longer-term approach allows each student to internalize the grade level material covered in their classroom and during test preparation sessions and firmly establish the building blocks to move towards mastery of above grade level material. Many of our students have commented that due to ISEE test preparation with My Learning Springboard faculty they feel better prepared for their school work and more confident when new material is introduced, especially in mathematics. This extended time frame for test preparation allows for true content mastery and creates a much better long-term value for students and families engaged in the process.

By Brad Hoffman, Board Certified Educational Planner and Learning Specialist, and Laurie Gross, Educational Consultant and Reading Specialist.

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