Posts Tagged ‘Teaching math’

What if… we did math right?

24 Oct

What if we made developmentally sequencel goals that we want our children to reach.

what if these goals includes procedures and concepts?

what if we designed learning material to help students achieve these goals and had multiple approaches to teaching allowing the child to work with the easiest approach for their learning style.

What if the students got lots of practice and teacher feedback.

what if students could pretest themselves to see if they need more time or are ready for a test.

what if we provide more time and resources when a child isn’t yet ready instead of just having then fail a test.

what if they take a test and we spend time afterwards, helping them correct and understand what they got wrong.

what if we don’t move on to the next unit until we have a certain level of mastery with the current unit.

what if we continue to include previous material in new units so students don’t forget older topics and retain their mastery.

why don’t we focus on quality over quantity and not worry about kids memorizing formulas but instead can they correctly use and apply them.

teachers can be guides and facilitators  in the classroom, kids who are further along can help give lessons to their peers, online resources can be used to free up the traditional approach.

Assessments can contain questions that require many different types of learning, procedural, conceptual, appplications, theorical, experimental, etc.

why can’t we change how we do math? It seems so simple.





Summer Math Preparation for the Common Core Student Grades 4-8

28 May

Recently I have been working with some students getting ready for their End of Grade Testing in North Carolina.  This is a couple years into our adoption of Common Core.  I have watched Common Core unfold and found both positive and negative things about the new standards.  This is not an article about those!  However, one of the big negatives is that students are spending more time on “conceptual” understanding and “alternative” (and long winded) approaches to mathematics that are getting further behind on the basics including the ability to do procedural mathematics.  In the mathematics education community, there has always been much discussion about the percentage of focus on procedural mathematics vs. conceptual mathematics.  Let me quickly summarize the current thought:

Procedural :  the ability to perform mathematics and solve actual problems such as 4X3 and 1/3 + 4/5.  Procedural mathematics gets a bad wrap, many educators assume that since calculators and computers can do this, our focus needs to be on the concepts behind the procedures and if we do this, the procedural math will come naturally.  This DOESN’T happen!  So many students can’t do the procedural math steps and we don’t spend enough time on them.

Conceptual:  the ability to understand the concepts surrounding the mathematics and the “why.”  One cannot solve a word problem involving multiplication if they don’t understand the concept of what multiplication is and how it is different from addition or subtraction.  However, one still needs to be able to do the procedural steps to get the answer to the problem and I disagree that the “procedural mathematics” will come automatically if a student knows the concepts.  Often, students only get to a partial understanding of the concepts and then they are in a real bind since they can’t do either the procedural or conceptual.

So, how does this apply to YOUR child?  Well, most children are just not given enough time practicing the procedures and this is what a parent must make up at home until these steps are automatic and easy.  Leave the conceptual teaching for the teachers (or get a tutor if needed for this) but you can do your child a huge service at no cost by having them master procedural mathematics!

This is not an inclusive list but a list of things that one should work on at each grade.  If your child is in 6th grade and hasn’t mastered the objectives listed for 4th grade, you need to start there.  Start at whatever level your child is not 95% successful at.

Grade 4:

  • Knows all multiplication and division facts (0-12)
  • Can do multi-digit multiplication
  • Can do long division with a single digit divisor
  • Can convert between mixed numbers and improper fractions

Grade 5:

  • Can add, subtract, multiply, and divide with fractions with unlike denominators (including 3 or 4 in a row)
  • Can add, subtract, multiply, and divide with mixed numbers
  • Can reduce fractions (including BIG fractions)
  • Understands divisibility rules and how to apply them (minimum can do 2, 5, 9, and 10)

Grade 6:

  • Can add, subtract, multiply, and divide with decimals
  • Can convert between decimals and fractions
  • Can solve 1 step algebraic equations
  • Can combine like terms in algebra
  • Can use distributive property in algebra
  • Can solve order of operation problems
  • Can find the 5 point summary for a box plot, graph, and interpret
  • Understands / memorizes appropriate math vocabulary:  mean, median, mode, range, IQR, Q1, Q3, variation, cluster, gap, outlier, MAD, standard deviation, spread, radius, diameter, etc.

Grade 7:

  • Can set up and solve ratio and proportion problems
  • Can set up and solve percent problems from words
  • Can solve 2 step algebraic equations
  • Can simplify algebraic expressions that mix distributive property and combing like terms
  • Can convert from words to algebraic expressions
  • Can solve 2 step algebraic equations that involve fractions and decimals
  • Can find volume and surface area of prisms
  • Can find area, perimeter, and circumference of mixed figures

Grade 8:

  • Can find slope from 2 points
  • Can find equation of a line from a slope and point
  • Can find equation of a line from 2 points
  • Can find equation of a line from a table or graph
  • Understands and can apply idea of x and y intercepts graphically and as points
  • Can find lines perpendicular to other lines
  • Can apply Pythagorean theorem
  • Can find the distance and midpoint between two points
  • Can solve more advanced algebraic equations including ones with fractions and decimals
  • Can solve problems involving the use of all exponent rules including negative exponents


I suggest that each student does 4 problems every night, this way they are not overloaded but they are doing math consistently.  They can’t move on until they are getting answers consistently correct and then always revisit old problems with new problems.  For example, your child might have mastered addition and subtraction of fractions with unlike denominators and they are working on multiplication and division of fractions now.  Some nights they will get all 4 problems of multiplication and division but some nights they will get a mix of old, one of each operation or even older material, one adding fractions, one dividing fractions, one long division (grade 4), and one reducing fractions (application to long division).

This IS the best way to help your child be successful.  Always check your child’s work after the first problem, you don’t want them doing all 4 problems wrong, then they are practicing how to do things wrong!  If it looks good, they can continue.  If they don’t get all answers correct, give them the first chance to fix it, if they can’t, then you can help them out or save up the pages and meet with a tutor and do these problems with a tutor.

Go to sites where they print out worksheets for your child and answers for you!  Sites such as math aids  and Kuta software.

Lynne Gregorio, Ph.D.

Mathematics Educator


Learning Mathematics Through Achievement Learning Based Model

07 Apr

What is an Achievement Learning Based Model in Mathematics?  This is a developmental model where the student has mastery of one topic before moving onto the next topic.  In using this model, one would first need to identify your academic goals.  For most educational institutions, the academic goals is that the student learn and maintain that learning.  Currently, most school systems use a model of a set of curriculum standards, currently the Common Core standards in much of the United States for Mathematics, and students are exposed to a classroom with objectives to teach these concepts that lead to the understanding and mastery of the standards.  Students start in Kindergarten and each school year is generally broken into 4 quarters where teachers are given a pacing guide so that they can get through all the objectives needed to FINISH the standards for each grade level.  Students are given classroom assessments along the way, including possibly quarterly benchmark tests and in later grades some type of End of Year test that measures the mastery of the standards for students.

The teachers are generally not allowed much freedom in deviating from this pacing guide and required topics that they must get through since these are all prerequisites for the knowledge base students will build on during the following year.  In the early elementary years, United States students actually do fairly well overall keeping up with the pace, although, there has been much controversy over the Common Core Standards for many different reasons including political reasons, lack of testing, and pedagogical methodology.  At some point, however, students (probably almost all students at one time or another) will come across a concept, unit, or topic either in elementary school, middle school, or high school where they will get confused.  There is no time built in for teachers to realize that the class is “lost” and they need to spend longer on a specific topic.  There isn’t even the flexibility for teachers to “speed up” when topics are easy so that they can slow down later when needed.  I have seen teachers finish a unit early and just give students “free time” since they were off their pacing guide.  Teachers are not being taught about “reading” students for understanding and “how to adjust” their lessons based on individual class feedback when taking methods classes because this isn’t even allowed anymore.  Teaching is no longer an art that allows for creativity and talent and therefore all the good teachers are leaving the profession in droves.

Let’s pretend for a minute that we care about student learning as our main goal.  If we consider an Achievement Learning Based Model, we can put student achievement before our need for control, before our need for cattle car education, before our need for convenience.  An Achievement Learning Model would require more work but in the age of technology, it is so very doable.  Many schools are considering a flipped curriculum these days.  A flipped curriculum if done correctly works like this:  lessons for each unit are taped, students watch the tapes for homework and then when they come to class, they spend classtime doing active learning with a teacher available for help.  Note, this is not time when the teacher sits and grades papers or takes time off, the teacher is actively participating with the students but it allows them time to have someone help with the active learning part, working problems rather than the static part of learning, watching the lesson.

Here is how the Achievement Learning Model can be added into the flipped curriculum model.  Let’s say that you have a group of high school freshmen who are taking Common Core Math 1 or Algebra 1 (we will just refer to to it as CCM1 here.)  Students will get a goal sheet of the units they need to cover, homework needed to turn in, and assessments they need to complete.  Students will watch the lesson at home.  The next day they come into the classroom and they work on problems assigned to them.  They start with easy problems and get problems that get more difficult as they are successful.  Once they are getting enough problems correct, they move onto to the second unit.  For some students this might be one day, for other students it might take longer.  When it gets close to quiz time, the student takes a practice quiz and self corrects the quiz.  The goal is that they don’t take an assessment until they are having success with their homework and practice quizzes.  If they did well (show mastery), they take their quiz, if not, they work more problems, get more help.  Each student works at his or her own pace.  However, the teacher does oversee the pace of each student and certain requirements are placed on students who are not putting in the effort (which is different from those struggling with the content).  At school, there are after school hours in place for students to come in and continue the same “work” they would do in class.  Every student will be successful since they don’t move on until they have shown success.  The goal is 4 years of math so students “take” math every semester, where a student ends up in their knowledge base will be different for every student.   At the end of CCM1, some will have finished the course and be ready to take the final exam.  If any finish early, they will be helpers to the remaining students!  What a great way to reinforce their knowledge.  If a student does not finish, they can continue in CCM1 the next year until it is complete and move into the CCM2 whenever they finish and start there.

Bright students may have a schedule that looks like this:

Block scheduling:  (just a sample)

First Semester        Second Semester

CCM 1                            CCM2

CCM3                             Precalculus

Calc AB                         Calc BC

AP Stat                          CCM – helper



A slower student might look like this:

First Semester           Second Semester

CCM1                               CCM1

CCM1                               CCM2

CCM2                               CCM2

CCM2                               CCM3

CCM3                               CCM3


Each student takes math every semester, each student has mastery but they get to do things at the pace they need and they will know far more mathematics than our current model where many get D’s and forget what they have learned.   This model needs the following to be successful:

1.  A good teacher who is excellent at explaining the content on the videos is easy to follow steps and includes problems for the students to “practice” while watching the video that shows that the student watched and paid attention to the video.

2.  A good curriculum writer who can create good practice problems so that students can have sufficient practice until they reach mastery with problems starting easy and getting more difficult and have practice quizzes and tests for students to take so that they know when they are ready for the real exam and ready to move forward.

3.  Teacher education where teachers are taught how to manage this new type of classroom, facilitate appropriate groupings among students working on the same topics, “read” students so they know who knows what and who is confused, be able to delegate helper students from within the class to students who need help, to be able to provide the best use of their time during the regular “workshop” settings of daily education.

4.  Test to see at what age students would be mature enough to handle “self” learning, although it will be new and a great skill that students will be learning so it is expected that students will have a normal adjustment period despite maturity issues.


Written by:


Lynne Gregorio, Ph.D.


New ways to help students be successful in mathematics

07 Nov

Although this idea applies at any age level, I am going to direct at the age I work most closely with – high school students.  Most of my students are North Carolina students taking the new Common Core 1, 2, and next year they will start Common Core 3.  I am slowly watching as the scope and sequence that North Carolina has chosen unfolds for each level.  Common Core 1 contains many Algebra 1 topics and Common Core 2 is more heavy on Algebra 2 than on Geometry.  I find that many of my students can do the work in isolation.  By this, I mean, I teach them a topic – for example solving exponent equations that do not require logs:  3^(x+1) * 9^(2x -3) = 9^(3x+4) and once I teach the topic, they get it and are able to do it!  They can complete a full page of work and get the answers right.  The next day, students will come in with another topic, maybe finding the inverse of functions.  I teach them the rule (exchange x and y, solve for the new y).  They then do a page of those problems correctly.  Prior to the test, students get many different topics in “isolation” from their teachers that we cover and they are successful at.  However, when the test comes, they fail!  Why?

Students don’t remember which problems require which steps.  They don’t spend time memorizing what type of problems match which type of problem solving skills and they seem to lack the ability to just look at a problem and use their overall knowledge of math (a long list of other math skills they have forgotten from previous years) to be able to reason out the answer.  Many times, they don’t even know what the question is asking. ” Oh, I did a page of problems where I switched x and y and then solved for the new y, but that was called an INVERSE?”

How do we fix this?  Well, it really needs to be fixed retroactively as students need to remember all their old skills as they move on and apply the new skills.  If one of their problems has a (1/81) and they are supposed to covert that to 3 to a power in Common Core 2, but they forgot all about negative exponents they learned in Common Core 1, then they are adding double the work.  They have the new skill to learn about solving the exponent equations and they have to re-learn all their exponent rules.  Hopefully, they are just “dusty,” and it doesn’t take too much and you can remind them that fractions mean negative exponents.

Moving away from the retroactive problem, let’s just focus on the best fix we can do.  When teachers assign homework, they assign a page of all problems from the current isolated area.  Solve 15 problems of exponent equations on Monday, solve 15 problems of inverses on Tuesday, solve 15 problems of solving radical equations on Wednesday, etc.  If in addition to this, teachers each night gave students a sheet with one question from each area to solve and the “wording” of how it will be asked of them – for example, in trig, we say, “Solve the triangle.”  What does that mean?  It means find all sides and angles, well if students don’t know, they will find out before the test!  Teachers forget that the goal is to help the students learn the material BEFORE we assess them!!

So, a student will be essentially be given “baby review” sheets all along and I wouldn’t even limit the questions to just the test questions since many of these students forget everything but if they do one problem each night from a section, it will keep them fresh.  How do I complete the square?  How do I find the vertex of a quadratic when it is in standard form?  How do I solve exponential equations?  How do I factor when the leading coefficient is not 1?  How do I factor difference of perfect squares?  Math asks you to remember a lot!!  We need to show kids how to do it!  We need to help them be successful.  If we can model good study habits, when they go to college, they will use these on their own.

If you are a parent with a struggling student, begin to make up (or hire a tutor) to make up baby review sheets that ask one problem of each type of thing the student should know.  When you see they are finally solid on a topic, remove it and just add it back randomly as a check!  Good luck!


Posted in Education


Review of Khan Academy

28 Jul

Okay, I am going to head over to Khan Academy and give you a blow by blow review from how user friendly it is to how well it teaches mathematics.

When I first get on the site, it clearly directs me where to go if I want to learn Algebra.   So I will click on the word Algebra.  However, after that first click, now I am bit more confused.  There is a place that says Introduction to Algebra so let’s start there.  Now I get some choices about history of Algebra and the Why’s of Algebra.  When reading another review, it commented about the lack WHY in Khan videos.  It appears that maybe they are trying to make up for some of that.  My problem is that this information needs to be integrated into each lesson not as a separate section.  It is like this is written here for an educator.  I don’t know that I see a student accessing the “Why we do things to both sides,” but let’s take a look.  After watching all these videos in this section, I give the overall videos a thumbs up!  However, the organization gets a thumbs down!  This should be considered a regular lesson for all of Khan academy for every topic.  For each topic, they should start with what they call the “WHY,”  I call it good teaching.  From there, they can move into some more procedural practice videos, and finally student practice videos.  The organization could look like this:

  1. Solving One Step Equations Lesson – here would be their “Why Videos” Relating to that
  2. Procedural videos for solving One Step Equations – this can be what I have traditionally seen on Khan Academy, many videos of people just solving problems
  3. Practice for the student (starting with easier problems and getting more difficult)
  4. NOW – Khan would move into the next developmental step
  5. Solving Two Step Equations Lesson – the rest of their WHY videos here.
  6. Procedural Videos for students to watch (starting with easier problems and getting more difficult)
  7. Practice for student (starting with easier problems and getting more difficult)

Let’s move on and see what else Khan has to offer…

So here is where things go down hill for Khan.  They just had these nice videos that did a good example of explaining the concept of solving one and two step equations.  Following down, the next thing on that page (which seems like what I should click on since it follows those videos) is something about Yoga… I decide to skip the Yoga and click on something that has some math terms in it so I click to the third section.  Here I get a list of options of videos to watch.  Now remember, I just learned some very basic ideas of how to solve an equation using a balance scale.  After getting confused by Yoga, I see Variables Expressions and Equations.  This is sort of random lecture that doesn’t really link me into anything and makes some medium jumps and has some confusing pieces in it.  The lesson is trying to teach you to substitute in a number to a variable expression but instead of building it in a developmental approach they jump to x+y+z=5 and start letting y=2 and z=3 and solving for x.  None of it is done in an organized fashion.  Letting that go, let’s just skip to the next video, maybe it will get better…

The next one says solving inequalities and equations through substitution (I picked #3).   Well, this involves solving equations, we did some of that but we have never talked about inequalities yet….  The problem says, “If r is the number of hot dogs Joey can eat in a minute, and N is the total number of hot dogs he eats in the contest, if Joey can eat 6 4/5 hot dogs per minute, how many hot dogs does he eat in the 10 minute contest?”  They give you the equation N/10 = r.

Here is the basic explanation they give to start, “He eats N hot dogs, and the contest is 10 minutes so we divide that by 10 and we get the number of hot dogs on average he eats.  So if Joey at 6 4/5 hot dogs per minute so they are saying r is 6 4/5, so they are saying, what is N going to be?”  Hmmm… did you follow that?  Did that explanation make you UNDERSTAND anything about the problem?  Do you get it better?  If the instructor just says, well plug in r= 6 4/5 and solve for N, have you learned anything?  One suggestion he gives to solve the problem is to “just try out numbers.”  Wow, that is a great idea (sarcasm).

Summary so far:

  1. You want to learn Algebra
  2. Go to site and find Algebra page – easy
  3. Great starter videos on how to solve one and two step equations
  4. After finishing those and maybe you can solve 2x -3 = 9 (by the way, only learned how to do it with numbers that work out)
  5. You see something about Yoga
  6. You skip Yoga and go to the next lesson and find problems like this hot dog problem that explains nothing, makes little sense, and is totally out of place.

Let’s leave behind “Introduction to Algebra” as that is clearly too hard and see if plain old “Algebra” is easier.

We choose linear equations, the first choice, and what is a variable, that sounds nice and basic.  Here we go, this is a much better place to start.  The lessons here so far have been basic and sequential.  A few minor problems are noted.  For example, when teaching how to plug in to evaluate an expression:  4n^1 + 2n^0.  (Carrot here means raised to the power of).  When “teaching” that anything to the first power is equal to itself, the instructor used a variable as their example x^1=x, this is teaching at a more abstract level rather than showing things such as 4^1 = 4 and 7^1 = 7 which is concrete and more easily grasped by the beginning student who just learned about x’s 5 minutes ago.

They begin to substitute in numbers with 2 variables and include positive and negative numbers.  In their example, they end up with -10 – 15.  They write the answer is -25.  The very first comment shows the lack of a strong teacher in these videos.  The first comment says, “why is -10-15 = -25, I think it should be 5.”  She continues to say, “Wouldn’t it have be -10–15 to get -25.”  Good teachers already know the mistakes that students will make and include this information into their lectures.  If I were to make this video, I would always quickly reteach older concepts whenever possible.  In this case, I would remind students that when dealing with positive and negative numbers and subtraction, you always want to do “switch change.”  Change the minus to a + and the sign of the number after that to its opposite.  -10-15 = -10 + -15, this makes seeing the -25 much easier for students.

Going back to their user interface.  One comment asked about practice problems.  The answer was that the lessons with the stars are the practice problems.  Having this written somewhere on the site would be helpful as it wasn’t clear to me either until I read the comment feedback.  On the practice problems, I do like the hint buttons and that you can ask for several hints if you get stuck.  However, I don’t like that the practice problems are not developmental.  The ideal situation would for them to start off easy and after you get 2 easy ones right in a row without hints, you get harder problems, and so on.

In the next unit, they talk about solving inequalities.  I don’t like the use of the words, “swap the inequalities.”  Most math teachers refer to it as flipping it.  The second thing is that their first introductory problem is -.5x < 7.5.  Their first step is to multiply by -2.  Again, “magic” math!  Most kids at this point CANNOT make the leap to seeing -.5 is =-1/2 and using its inverse.  Two seems like an odd number to throw into a problem with 5’s and 7.5’s.  Don’t start with something like that when the point you are trying to make is about flipping inequality signs, why muddy the water with something so confusing as -.5 and its inverse being -2.  He also immediately introduces the idea of how to graph inequalities on a number line and use interval notation.  He just sort of does it, no explanation of the number line and a quick explanation of the interval notation.  After years and years of experience with students, I know that kids can’t just hear that or view it at that speed with such little discussion and explanation and understand the concept.  There needs to be a whole lesson just on graphing inequalities (when to use open and closed circles and why) and writing interval notation, not an after thought in this lesson.  This is also another issue with Kahn.  Some areas they go so slow and other areas that need to go slow, they whip through it so fast.  It is like they have never worked with real students of various levels before.

We are at the point where big leaps are being made in Algebra.  I was pleased with the intro to solving equations videos.  I liked the discussion about what a variable and overall most of the plugging in to equations by substitution wasn’t too bad but now that we are getting into the meat of Algebra, we will start losing students.

At this point, I randomly picked another topic to see how it was done.  I picked completing the square.  If you read the comments you get the full the picture:

  1. I don’t understand why c=22
  2. I can’t see how this video has anything to do with practice
  3. Is this Algebra 1?
  4. I still don’t get how to do this?  Why is the FOIL method on the right?
  5. I don’t get that pattern about (x+a), my teacher showed us a different way

The teacher goes through a symbolic proof.  Although this is nice for Algebra teachers, be real, the majority of kids don’t get this at this level.  What you need to do for kids is provide a written set of steps, let them know where it is used, why it is important, and situations of application.

Overall, there are some decent videos but the entire system needs an overhaul.  Here are my recommendations.

  1. Someone who is a very strong math educator and who really knows how to teach well should go through each video and make suggestions for change.
  2. The organization needs updating for ease of use.
  3. A developmental approach needs to be used for both problems in lessons and practice.
  4. Teachers should not rely on this for their lessons as it stands, the lessons are not strong enough and it is missing the benefits of dynamic teaching.

Written by:  Lynne Gregorio, Ph.D. Mathematics Education