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Problem-Solving Strategies

January 26, 2021 Brad Hoffman Leave a Comment

$Math-Manipulatives$ 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

Educational Planning With A Multidisciplinary Team

January 3, 2021 Brad Hoffman Leave a Comment

January is always an important check-in time for our families with parent-teacher conferences having recently occurred and new year’s goal setting top of mind. Most schools have sent the first set of report cards and/or progress reports by now, and those who haven’t will send them this month. In addition, many families have completed or are working to complete the admissions process for this past fall, while other families are starting to plan for the admissions process for the upcoming fall, whether they’re applying for the 2s, preschool, kindergarten, 6th grade, 9th grade, college, graduate school or something in between. Because families are faced with so many considerations and decisions, there is a tremendous advantage to educational planning with a multidisciplinary team.

At this time of year, there are many important questions that our families are discussing with us, including but not limited to:

My preschooler will just miss the cutoff for starting Kindergarten this upcoming fall, what should we do?
How do we manage private school placement if we’re moving to NYC from a foreign country or another state?
How do we change schools if we’re not totally happy in our current school?
What is my child’s reading level and how does it compare with quarterly benchmarking and grade level expectations?
How do I help my child to master their basic facts? Improve math fluency?
How do I manage accommodations for standardized testing, including extended time, typing an essay, or marking directly in the test booklet?
Is a neuropsychological evaluation valuable?
Should I take the ACT or SAT and when should I start preparation? How do we juggle the demands of junior year?
Should we consider boarding schools? If so, why?
What is Singapore Math? Everyday Math? Exeter Math?
How can we enhance executive function skills within the context of the school curriculum and content support?

As a multidisciplinary educational consulting firm focused on supporting the entire educational journey, our executive team and faculty has had the privilege of working with so many great families to accomplish so many targeted goals over our 15 years as an organization, and our hundreds of years of combined professional experience. In each instance, we define success in partnership with families and work toward that shared goal.

There are myriad scenarios that occur for families over the course of the school year and our team is here to help with educational planning through an integrated and collaborative approach. We know results are best achieved when we work as a team in partnership with parents and schools, and we believe in customizing the approach for each of our students and families in order to create an optimal experience.

Please contact our office to discuss your needs.

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

Under the leadership of Brad Hoffman and Faya Hoffman, a husband-and-wife team, My Learning Springboard offers private tutoring, test preparation, special education advocacy, admissions consulting and school transition support at every level from nursery through college, including specialized school placements for students with learning disabilities and therapeutic school settings.

In addition to their years of classroom teaching experience and consultation with both families and schools, Brad Hoffman is a Professional Member of the Independent Educational Consultants Association (IECA) with speciality designations in mainstream (day and boarding schools), specialized (learning disabilities), and therapeutic school placements. He is also a member of The Association of Boarding Schools (TABS), The Enrollment Management Association (formerly SSATB), the National Association for College Admission Counseling (NACAC), the National Association of Therapeutic Schools and Programs (NATSAP), the Association of Educational Therapists (AET), the Learning Disabilities Association of America (LDA), and the National Council of Teachers of Mathematics (NCTM). My Learning Springboard is a member organization of the Washington Independent Services for Educational Resources (WISER).

Why Studying Grammar Matters

November 13, 2020 alla Leave a Comment

In an age of auto-correct and casual texting, it’s common to think of grammar as a rigid set of rules that is unnecessary to study. Often, students have an intuitive grasp of language and what “sounds right” based on years of reading and writing. Educators face similar questions about studying grammar. In fact, debates over teaching grammar have been raging in the United States for the last fifty years. Those who advocate against teaching grammar argue that students will learn grammar in an organic way, through reading and writing. They state that grammar instruction is often prescriptive and focused on rules, thereby limiting student expression. Certainly, there are dreary and constricting ways of teaching and learning grammar, but can we ever separate our thoughts and ideas from the sentences we are using to express them?

When we understand the relationship between words in a sentence, we gain a clearer sense of what it is that we mean to say. Rather than seeing grammar as a “mechanical” layer of writing, we can use grammar to more fully understand the linguistic choices we’re making and their effect on our readers.

Here we’ll take a look at some of the ways studying grammar can help our writing and language acquisition.

Fluency

When we understand the necessary parts of a sentence, we can feel confident about clarifying convoluted phrasing. Let’s look at an example:

June was at the grocery store when she ran into a girl who used to go to her school named Essie, and she bought three avocados and a seltzer.

This sentence is technically correct but reads quite sloppily. We aren’t sure who bought the avocados and seltzer, June or Essie.

One way to clarify the sentence is using a prepositional phrase to situate us in place, adding an appositive to separate the additional details about Essie, and adding in the adverb “then” to clarify the order of actions.

Syntax

Often, complex syntax can reflect a nuanced and sophisticated way of connecting ideas. By studying grammar, we can start to make our sentences more intricate, and more aligned with our critical thinking.

For example, let’s say we are thinking about the benefits and drawbacks of remote learning.

On a first draft, we might write: Remote learning is beneficial for students who can focus better in their own homes. It is harmful to those who don’t have access to technology or who focus better in a group setting.

Then, we decide that we want to directly show the relationship between the benefits and drawbacks and can use transitional words to combine the sentences.

While remote learning is beneficial for students who can focus better in their own homes, it is harmful for those who don’t have access to technology or who focus better in a group setting.

Remote learning is beneficial for students who can focus better in their own homes; however, it is harmful for those who don’t have access to technology or who focus better in a group setting.

Varying syntax also helps the reader stay engaged with your writing. When our language is monotonous, the reader may have a hard time moving through a piece of writing. Even when reading silently, our “inner ear” enjoys musicality, or at least a change of pace between sentences. Studying grammar can open up more choices for us to create dynamic sentences.

Here is another example:

I walked my dog and I got some flowers. I read a book and I called a friend. I took a shower and listened to music before bed.

All of the sentences above use the simple past verb tense. The syntax does not vary and as a result, the sentences sound more like a list of activities than a narrative.

In the revised sentences, both simple past and past continuous verb tenses are used, which makes the day flow in a livelier manner. By adding in prepositional phrases are added, we are not only mixing up the syntax, but also conveying important details. Adverbs help to locate the activity in time.

Learning another language

Another reason to study English grammar is to improve our ability to learn another language. Students often find that having a familiarity with parts of speech, tenses, and other grammatical components can help them transition into learning other languages. Likewise, students often find that studying another language improves their understanding of English grammar. Even when the grammar structure varies dramatically between two languages, understanding these differences can help students avoid common errors. There are many similarities in grammar structure between English and Mandarin. However, English-speaking students studying Mandarin often face difficulty with adverbs and denoting amount. In order to learn Mandarin grammar, it’s helpful to understand where adverbs are placed in English and how “measure words” are used. Once we start studying grammar across multiple languages, we begin to see how the language choices we make are not arbitrary. Rather, grammar is the architecture of language, setting up the foundations – and possibilities – for how we can express meaning.

Where to Begin

The good news about studying grammar is that we can find it all around us. When reading an interesting article, we can try to identify verb tenses or ways that the syntax is varying. When reading a poem, we can consider the grammatical choices an author made, and how they affect the text. Even when casually texting, we can think about the linguistic choices we are making.

For additional resources, Purdue Owl has excellent online resources on grammar and writing. The classic book Elements of Style by E.B. White and William Strunk, Jr. is a helpful guide to writing clearly and concisely. Additionally, The Blue Book of Grammar and Punctuation by Jane Straus offers clear instruction on grammar and punctuation usage, as well as detailed examples and quizzes.

By Joss Lake, MFA | Private Tutor and Writing Coach

The Science of Scientific Studying

November 8, 2020 Brad Hoffman Leave a Comment

So, you are tutoring a student in science and they are having trouble with a particular concept- to be a successful teacher you have to have a variety of different approaches on hand for moments like this. If your student is having trouble getting a grasp on tough biology or chemistry topics, you have to figure out how to re-present the information in a new way that might work better for the way your student learns. Here are five simple tips to revamp the way you approach science concepts as an instructor- they will also help make the information stick long after the test:

1) Make a story out of it!

Science textbooks are full of jargon that can be not only incomprehensible but also uninteresting. Students can be turned off from learning about a topic that seems too complex or boring because of poor presentation. The solution? Encourage your student to devise stories with characters, settings and plots based on the material at hand. The more creative, detail-specific, and relevant to your child’s life and interests the story can be, the more likely he or she will ultimately grasp the information. As teacher and MIT Ph. D. candidate Tyler Dewitt points out in this TED Talk, crafting a narrative around a difficult topic can make it easier to understand as well as much more interesting for your student. The act of narrative generation pushes your child to process and understand the material to a higher level that simply reading a textbook does not achieve on its own.

2) Map it out

Another way to re-present challenging information to facilitate learning is via mind maps. Mind maps are one way to outline information in diagram form that emphasizes connections between topics. Although simple in concept, they can be a powerful way to reorganize information that may be otherwise tedious to remember. Check out this YouTube clip for more information and suggestions on how to begin, but as with study strategies in general, your student should experiment with different methods to find out what works best. A variety of free mind mapping software can also be found online.

3) Note your questions

Take note of how your student takes notes. Does he or she write text verbatim? Are there bullet points, stars, or drawings? While any number of styles might work, one helpful strategy might be to help your child create a structured way of recording questions during class or while studying. Highlighting what we don’t know can be as helpful later on as marking down what we do. Designate a portion of a notebook or select a specific font that stands out. The final and most important part? Look up the answers later on!

4) Why is it called that?

Writing down words that are unfamiliar and looking up their definition is always a good idea. Looking up the origins or roots of a new phrase can be equally powerful. For example, consider this information about the heart. The heart is effectively a muscular pump, and like every other muscle of the body, it needs the oxygen and nutrients found in oxygenated blood to function. It receives its supply of blood from the coronary arteries. Without any additional knowledge, making the arbitrary association between “coronary arteries” and “blood supply to the heart” might be a stretch on a tricky multiple choice question or a diagram of various arteries in the body. Thankfully, the word “coronary” is there to help out. The word coronary is derived from a Latin phrase that means wreath or crown, and the coronary arteries wrap around the top of the heart like their namesake. Having this extra piece of information makes the association between the arteries’ name and function less arbitrary by alluding to the location and path the coronary arteries follow. Taking a few seconds to look up the roots behind other difficult biological phrases will cut down on the amount of strict memorization and will aid in fact recall during assessments.

5) Bit by bit

While cramming for exams may be unavoidable at times, learning how to master challenging topics through frequent study is instrumental to success in the long run. Help your student generate a routine to work on the subject or topic they struggle with the most (the very one they may be avoiding) for a little bit each day. Set aside about thirty minutes each day to review the content of that day’s class and to map out how it fits into the broader scheme of the course – and then start on the associated homework. If possible, previewing material for the next day of class by looking ahead at class notes or relevant pages in the textbook can help your student get a jump on the material. The point is to avoid those cram sessions and empower your student to construct their own study routine – your student will thank you when they get to college!

Mix and match these strategies to see what works best for your student. Ultimately, the goal of these tips is to inculcate simple strategies for proactive learning that will assist with your science classes now and in the future!

Written by: Editorial Team, My Learning Springboard, Inc.

In Defense of Re-Reading

November 5, 2020 Brad Hoffman Leave a Comment

“Serene was a word you could put to Brooklyn, New York. Especially in the summer of 1912. Somber, as a word, was better. But it did not apply to Williamsburg, Brooklyn. Prairie was lovely and Shenandoah had a beautiful sound, but you couldn’t fit those words into Brooklyn. Serene was the only word for it; especially on a Saturday afternoon in summer.”

For at least four years in a row, sixth grade through ninth grade, I opened my school year by reading these words, the opening lines of Betty Smith’s coming-of-age novel A Tree Grows in Brooklyn. As soon as the library was open to students, I would be there, heading for the “S” shelves of the fiction section. Over the years, I would reread many other books as well, such as Wuthering Heights by Emily Bronte, Jane Eyre by Charlotte Bronte, and The True Confessions of Charlotte Doyle by Avi. It wasn’t that I didn’t like reading new books. In fact, I was a voracious reader, and most of the books I read were new to me. There was a comfort in revisiting those favorite stories, though; they were old friends I could always rely on. Looking back, I don’t think it was by chance that I always chose to start the year with a book that I knew I would enjoy.

When I help students choose books, I try to keep in mind my own rereading habits. As educators and parents, we naturally want children to learn new things. Generally, this means that we want them to read books that they have never read before. When we see a child selecting independent reading materials and we notice them reaching for a book that we know they have read, our natural instinct is to redirect them to a new one.

We might want to rethink that instinct, however. A recent study has found that rereading a favorite book benefits the reader by promoting self-reflection, enhancing emotional health and facilitating a deeper understanding of the text. It’s that deeper understanding that allows for the sort of textual analysis that students perform on texts in secondary and college classrooms, so rereading, although I wouldn’t recommend it as part of a curriculum, serves a definite academic purpose. On a more basic level, rereading allows for better comprehension of the sort that comes along with more life experience and sharpened reading skills. I definitely understood Great Expectations and A Tale of Two Cities better when I revisited them in high school than when I first read them as an ambitious middle-schooler.

Could there be such a thing as too much rereading? Undoubtedly. Young readers should be reading far more new texts than old ones. It is possible that a young reader might get “stuck”in a group of favorite books, particularly a story series, and refuse to move forward. It’s a rare phenomenon, though, and usually a temporary problem —one that can generally be fixed by nudging the student towards books with similar themes.

An occasional rereading experience serves one of the highest purposes of education: By empowering young readers to amass a personal literary canon, it encourages them to become lifelong readers. It has been an extraordinary experience for me as a teacher to revisit some of the young adult novels that I’d last read as a child —Maniac Magee by Jerry Spinelli, I Am the Cheese by Robert Cormier, The Pigman by Paul Zindel. Turning over the pages, I thought about the girl in the middle-school library who had pulled them off the shelves so many years ago.

Maybe that’s the real magic in rereading —the years pass, the world changes, but the books are always reliable. Maybe the word “rereading”is a misnomer. As Verlyn Klinkenbourg wrote in the New York Times, “The real secret of re-reading is simply this: It is impossible. The characters remain the same, and the words never change, but the reader always does.”

By Elizabeth Walters, Private Tutor

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