david stevens, ed.d
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DESCRIPTIONHow Math Learning Difficulties Develop. David Stevens, Ed.D. Developmental Psychologist. 01 Developmental Readiness. Some students arrive in kindergarten and first grade without having the necessary foundation for math instruction. Robbie Case & Sharon Griffin. Number Knowledge Test. - PowerPoint PPT Presentation
David Stevens, Ed.D.Developmental Psychologist
How Math LearningDifficulties DevelopThe purpose of this presentation is to illustrate how some students develop math learning difficulties. Over the past decade research studies have documented how specific teaching techniques and curriculum decisions have not provided our students with the mathematical foundation they need to succeed. The presentation will also provide research-based solutions to these difficulties.
01 Developmental ReadinessSome students arrive inkindergarten and first grade without having the necessary foundation for math instruction.
Robbie Case & Sharon GriffinSome students arrive in kindergarten and first grade without having the necessary mathematical foundation to benefit from grade level instruction. For a variety of reasons some students have not developed the most basic understandings of quantity and number by the time they begin formal schooling.
Number Knowledge Test
Have a candy. Here are 2 more. How many do you have?Which pile has more?(Show two piles of chips)How many triangles are there?(Show mixed array)If you had 4 candies and received 3 more, how many would you have?What comes two numbers after 7?Which number is bigger/smaller? (Show two arabic digits)Robbie Case & Sharon GriffinIn order to better understand how some students begin school ready for math instruction and others do not, two researchers designed a study to evaluate whether students beginning kindergarten had the necessary mathematical foundation. Robbie Case & Sharon Griffin interviewed students entering kindergarten. They asked them the following questions.
They analyzed the data by two groups. The first group they considered not at risk for math failure. The second group they considered at risk for math failure because they came from lower income households, and research has shown that students from lower income households often start behind in math and never catch up . . . generally speaking.
(Click to bring up data from not at risk group green bars)
The green bars show the passing rate for the Not at Risk group on the various questions. Generally the group did pretty well.
(Click to bring up data from At Risk group red bars)
Here is the data from the group that is at risk for math failure. They did not do as well, but on the first three questions did nearly as well. However, it is on the last three questions that there is a big gap in the passing rate and really shows the different levels of understanding that the two groups have of foundational math concepts.
Now, generally speaking, both groups typically receive the same math instruction based on worksheets, writing numbers, and adding and subtracting. But clearly one group is ready for this type of instruction while the other is not. The at Risk group does not understand, for example, that the numeral 7 represents a larger quantity than the numeral 4. This group needs a different type of instruction and curriculum than the group that understands this concept of quantity.
02 Mile Wide Inch ThickThe large number of required topics does not allow some students the time they need to understand foundational concepts.
Schmidt, McKnight, & Raizen 1997Research in the late 1990s compared the US math curriculum to that of the countries who consistently out performed the US on international tests of mathematical ability. The researchers concluded that the US curriculum was overly broad, with teachers required to teach many different topics which did not leave time for in-depth learning and understanding.
This slide shows which topics the top achieving countries teach in grades 1-8. Notice that in grades one and two the top achieving countries only teach three topics. This allows them to go into more depth and ensure their students have mastered these foundational concepts.
This slide shows the topics that need to be covered in each grade (1-8) according to the 1989 NCTM Standards. Nearly every state adopted the 1989 NCTM Standards as the framework for their own standards. Many students have their own variations, but for the 90s and 2000s most states were using this type of framework.
GradeNumber of Topics per GradeCenter for Research in Math & Science Education, Michigan State UniversityThis graph compares the number of topics covered in the US (in most states) to the number of topics covered by the top achieving countries. This slide is just another way of looking at the data from the three previous slides.
Center for Research in Math & Science Education, Michigan State UniversityPercent CorrectThere are 600 balls in a box, and 1/3 of the balls are red.
How many red balls are in the box?Grade 4 International Test QuestionThis fraction problem appeared on the 4th grade international test. 80% of the top achieving countries students answered this question correctly. Only 38% of US students answered this question correctly.
What is interesting is that in the US we start teaching fractions in kindergarten and first grade. The top achieving countries dont start teaching fractions until 3rd or 4th grade. Despite having started learning fractions 2-3 year earlier the US students did not do nearly as well on this fractions problem.
03 Teacher Training ProgramsTeacher training programs traditionally emphasize Language Arts instruction.
Another factor in this discussion is the way that we teach math. Typically, teacher education programs emphasize Language Arts instruction. Schools emphasize reading over math. This leads to math instruction that often is influenced more by reading principles than math principles. It is important to remember that there are significant differences between math and reading.
Colleges Providing Sufficient Trainingwww.nctq.org
One study of teacher training schools found that only 13% of the programs provided sufficient training for elementary math instruction.
04 Memorization over UnderstandingSome students move forward in the early grades only by using counting and memorization strategies.
A combination of these factors lead many students to progress through the math curriculum in the early grades mostly through memorization.
How Many cookies?Which is larger 8 or 9?5 + 3 = ?4 x 6 = ? 328 + 486
Here are some examples of problems that we can solve through memorization and counting strategies and arrive at the correct answer . . . even if we do not understand its meaning.
When a kindergarten student is asked How many cookies are on the plate?, that student counts all of the cookies. She may not understand what how many means, or what the concept of quantity is. But she has learned that when the teacher says how many, she should count the objects.
When a kindergarten student is asked which number is larger, 8 or 9, that student answers 9, because he has learned that the number that comes later in the counting sequence gets the right answer. He does not truly appreciate that numerals represent quantities, that 9 is slightly larger than 8 and much larger than 2. But he has been told that the number that comes later is larger and the earlier number is smaller.
In grade one a student sees a worksheet problem, 5 + 3 = ?. The student counts to 5, then counts on 3 more. She might even make five little dots on the 5 and three little dots on the three and then count them all. She arrives at 8 as her answer. But she might not understand that she has taken two parts and combined them to make a larger whole.
In grade two a student sees a problem like 4 x 6 = ?. This student has learned to skip count by sixes. He counts 6, 12, 18, 24. Or maybe, he has memorized his multiplication facts and can recall that the answer is 24. But he does not appreciate that he has created 4 equal groups of 6.
In grade three a student is confronted with 328 + 486 = ?. She solves 8+6 in the ones column, carries the one and solves 8+2+1 in the tens column, carries the one and solves 4+3+1 in the hundreds column. She produces the correct answer, but does not recognize that the one she carried the first time is actually a ten, and that the second one she carried is a hundred. She is simply manipulating numerals and following procedures. If she had arrived at 114 as the answer instead of 814 she might not even notice anything wrong because she is not really thinking about what the numbers mean.
2 + 3 = ?3 + 2 = ?2 + ? = 5 5 = 3 + ?5 - 3 = ?5 - 2 =?
This slide shows a series of number relationships. A student with an in-depth understanding of the part-to-whole model can recognize the relationship between these number sentences. Once she figures out that 2+3 is five, she also knows that 3+2 also equals five. Similarly, she knows that the missing number in 2 + ? = 5 is 3. If we move the equals sign to the left side of the number sentence and present the sum first this does not throw her, because she understands the equals describes a balance between the two sides of the equation. Since she understands the missing addend problem (5 = 3 + ?) she can make the connection to subtraction and knows that 5-3=2 and 5-2=3. The students understands all of these problems as variations of the same part-to-whole problem.
05 Curriculum Becomes More ComplexAs the curriculum becomes more complex the counting and memorization strategies are not effective.
As the curriculum becomes more complex, typically in 4th and 5th grades, the counting and memorization strategies that produced correct answers in the primary grades are no longer effective. Topics such as fractions, decimals, ratios and long division do lend themselves as well to simple counting on and counting back strategi