By Wendy Dubas, Lower School Math
International Math Assessments
The workshop began with a discussion on the Trends in International Mathematics and Science Study (TIMSS), which is an international assessment that basically involves number crunching. In 2007, 36 countries participated in the TIMSS, and the average score was 500. American 4th graders had an average score of 529 and ranked 11th among the 36 countries. American 8th graders had an average score of 508 and ranked 9th.
In 2009, 15-year olds from 65 participating countries took the Program for International Student Assessment (PISA). The PISA differs from the TIMSS as it tests on how well students can apply math. In 2009, the average U.S. score was 487 which was lower than the international average score of 496. The U.S. ranked 31st out of the 65 participating countries.
In both the TIMSS and the PISA, Chinese Taipai, Korea, Singapore, Hong Kong, and Japan were ranked at the top. A discussion followed on factors that contributed to the high scores of these Asian countries and the lower U.S. scores:
• Part of the score discrepancy is cultural. U.S. children spend around 180 days in school. In South Korea, children spend 220 days in school and children in Japan spend 243 days in school. Their school day is 10-11 hours. This totals to three more months of school than in the U.S
• The presenter, Feifer, states that “the language of math matters”. Many of the Asian countries have verbal lingo that is a base-ten counting language. For example, in Chinese the language for the number eleven would mean 1 ten and 1 one. This is important in efficiency in mathematical problem solving. In the U.S. system, however, the words for the numbers 11-19 don’t provide immediate place value connection or understanding. Also, Chinese numbers are brief which is helpful for more efficient memory. By the age of 4, many Chinese students can count to 40 whereas most U.S. students can count to 15.
• Math skill building in the lower grades needs to be fun and opportunities to practice skills with math games and activities should be provided.
• U.S. classes often focus too much on answers. Students should practice multiple methods of problem solving.
• Instruction is often only 45 minutes per day and often in the afternoon.
Feifer gave an example of what he considers to be a good math test, which would give the student the problem and multiple answers. Students need to identify which answers are incorrect and why.
Math Disabilities and the Brain
Students with math disabilities are slower with basic math processing skills such as identifying numbers, giving a quantitative value to a number, comparing numbers, and counting forwards and backwards. Children also struggle with visualizing numbers, seeing patterns, using strategies, and remembering steps in algorithms. They frequently need help getting started with a math problem.
Children with math disabilities have considerably weak skills in computation and cognitive processes used to solve math problems. This includes difficulties with:
1. Poor language and verbal retrieval skills (temporal lobes of the brain); Feifer states that “early math skills tend to be verbally encoded.” To solve word problems, for example, children need both math and language skills. Words such as some, sum, together, difference more than, fewer, etc. can be confusing for children with math disabilities.
2. Poor Working memory skills; remembering previously learned information and learning new information requires working memory. Working memory keeps information active and alive. There are three areas of working memory:
• Phonological Loop (temporal lobe of the brain): This holds and uses auditory math language. The phonological loop helps in the retrieval of math facts and the writing of numbers that are dictated aloud to the student.
• Visual-Spatial Sketchpad (parietal lobes of the brain): This area of working memory allows children to use mental imagery to temporarily store visual/spatial information. The visual-spatial sketch pad helps children to visualize numbers and is used in mental math and comparing numbers. When children have a weak visual-spatial sketch pad, they can externalize their working memory by counting on their fingers, for example.
• Central Executive System (frontal lobes of the brain): This area of working memory regulates anxiety and distracting thoughts. Anxiety impacts working memory.
There are several interventions for difficulties in working memory:
• Providing practice estimating the number of marbles in a clear jar helps students think in pictures.
• When children have difficulty using mental imagery to store visual/spatial information, externalize the working memory by having students use manipulatives to count.
• Have a number line on the student’s desk
• Set a tone in the classroom where children know it is a safe learning environment. This will help reduce anxiety and encourage risk taking.
• Present more than one approach to problem solving and encourage student chosen algorithms. This will help with the student’s working memory.
• Teach skip-counting to learn multiplication facts.
• Have students explain their strategies when problem solving.
• Develop number sense through math games and activities.
3. Poor Executive Functioning Skills (frontal lobe); Feifer defines executive functioning skills as “a set of directive processes such as planning, self-monitoring, organizing, and allocating attention resources to effectively execute a goal directed task.” He states that the key to math success is executive function. With good executive functioning skills, a student can stay focused, allocate time appropriately, learn from feedback to try a different approach when making an error, ask good questions, and have organized thought processes. Feifer states that when executive dysfunction occurs, the following math skills are affected:
• Selective Attention
• Difficulty following algorithms.
• Inattention to operation signs.
• Planning Skills
• Difficulty determining important information in word problems.
• Selecting a math process is difficult.
• Organization Skills
• Challenges in setting up problems.
• Errors in lining up math equations.
• Does not check work.
• Unaware if answer makes sense.
Some useful interventions would be to have students use graph paper to line up equations, and to allow students to select their own algorithms.
It was interesting to learn why children make certain math errors. Here are some examples taken from Feifer’s handout:
Math Fact Error
6 + 5 = 10 Verbal Retrieval
Operation Error 6 – 5 = 11 Poor attention or executive function.
Algorithm Error 123
44 Poor working memory in applying directionality.
Place Value Error .70
Poor working memory in following the procedure.
- I have a better understanding of how important it is to teach different strategies. I have talked to students about how each of us learns math differently, so some strategies work better for some than others. I learned that when students can choose the strategy that works better for them, this helps the student’s working memory.
- One of my first grade students has a highly anxious personality, and she struggles in math. I have a better understanding of how this anxiety can affect working memory. Although I have always tried to have a relaxed and comfortable environment in math, since the workshop I am especially conscientious of its importance for this particular student.
- With my better understanding of working memory, I see this may be one of the reasons that several first grade students struggle with math. To help with the visual/spatial sketchpad, I sent home manipulatives (unifix cubes) with a few students to use while doing homework.
- Generally, I use this knowledge and understanding of how executive function, language, and working memory affect how students learn and retain math skills in how I teach. For example, I continue to use manipulatives to introduce new math skills, children practice math skills with activities, second graders are taught to draw lines dividing their ones, tens, and hundreds place to keep their columns straight, etc.)